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- .gitattributes +170 -0
- parse/train/1-j4VLSHApJ/1-j4VLSHApJ_layout.pdf +3 -0
- parse/train/1-j4VLSHApJ/1-j4VLSHApJ_origin.pdf +3 -0
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- parse/train/BJE-4xW0W/BJE-4xW0W_middle.json +0 -0
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- parse/train/BJbD_Pqlg/BJbD_Pqlg_middle.json +0 -0
- parse/train/BJbD_Pqlg/BJbD_Pqlg_model.json +0 -0
- parse/train/BJeWUs05KQ/BJeWUs05KQ_layout.pdf +3 -0
- parse/train/BJeWUs05KQ/BJeWUs05KQ_origin.pdf +3 -0
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- parse/train/BJxH22EKPS/BJxH22EKPS_layout.pdf +3 -0
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- parse/train/BkgnhTEtDS/BkgnhTEtDS.md +381 -0
- parse/train/BkgnhTEtDS/BkgnhTEtDS_content_list.json +0 -0
- parse/train/BkgnhTEtDS/BkgnhTEtDS_middle.json +0 -0
- parse/train/BkgnhTEtDS/BkgnhTEtDS_model.json +0 -0
- parse/train/BklIxyHKDr/BklIxyHKDr_layout.pdf +3 -0
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- parse/train/ByG8A7cee/ByG8A7cee.md +331 -0
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- parse/train/ByG8A7cee/ByG8A7cee_model.json +0 -0
- parse/train/ByJWeR1AW/ByJWeR1AW_layout.pdf +3 -0
- parse/train/ByJWeR1AW/ByJWeR1AW_origin.pdf +3 -0
- parse/train/ByJWeR1AW/ByJWeR1AW_span.pdf +3 -0
- parse/train/CBmJwzneppz/CBmJwzneppz_layout.pdf +3 -0
- parse/train/CBmJwzneppz/CBmJwzneppz_origin.pdf +3 -0
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- parse/train/Ggx8fbKZ1-D/Ggx8fbKZ1-D_layout.pdf +3 -0
- parse/train/Ggx8fbKZ1-D/Ggx8fbKZ1-D_origin.pdf +3 -0
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parse/train/rkxJus0cFX/rkxJus0cFX_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/kWSeGEeHvF8/kWSeGEeHvF8_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rkxJus0cFX/rkxJus0cFX_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rkxJus0cFX/rkxJus0cFX_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/kWSeGEeHvF8/kWSeGEeHvF8_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/kWSeGEeHvF8/kWSeGEeHvF8_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/kWSeGEeHvF8/kWSeGEeHvF8_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rkgKBhA5Y7/rkgKBhA5Y7_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rkgKBhA5Y7/rkgKBhA5Y7_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rkgKBhA5Y7/rkgKBhA5Y7_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/BklIxyHKDr/BklIxyHKDr_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/BklIxyHKDr/BklIxyHKDr_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r1xMH1BtvB/r1xMH1BtvB_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r1xMH1BtvB/r1xMH1BtvB_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/SJlbyCNtPr/SJlbyCNtPr_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/LVWcGZr-8h/LVWcGZr-8h_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/LVWcGZr-8h/LVWcGZr-8h_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/LVWcGZr-8h/LVWcGZr-8h_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/HyEtjoCqFX/HyEtjoCqFX_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/SJZAb5cel/SJZAb5cel_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/H1TWfmnNf/H1TWfmnNf_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/Hk4dFjR5K7/Hk4dFjR5K7_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/Hk4dFjR5K7/Hk4dFjR5K7_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/Hk4dFjR5K7/Hk4dFjR5K7_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r1lL4a4tDB/r1lL4a4tDB_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r1lL4a4tDB/r1lL4a4tDB_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r1lL4a4tDB/r1lL4a4tDB_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/Ggx8fbKZ1-D/Ggx8fbKZ1-D_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/Ggx8fbKZ1-D/Ggx8fbKZ1-D_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/CBmJwzneppz/CBmJwzneppz_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/CBmJwzneppz/CBmJwzneppz_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/CBmJwzneppz/CBmJwzneppz_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/HkzZBi0cFQ/HkzZBi0cFQ_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/HkzZBi0cFQ/HkzZBi0cFQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/HkzZBi0cFQ/HkzZBi0cFQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rJeZS3RcYm/rJeZS3RcYm_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/rJeZS3RcYm/rJeZS3RcYm_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/ryxepo0cFX/ryxepo0cFX_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/ryxepo0cFX/ryxepo0cFX_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/ryxepo0cFX/ryxepo0cFX_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/dUk5Foj5CLf/dUk5Foj5CLf_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/dUk5Foj5CLf/dUk5Foj5CLf_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/dUk5Foj5CLf/dUk5Foj5CLf_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r6cNUjS8cm0/r6cNUjS8cm0_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r6cNUjS8cm0/r6cNUjS8cm0_layout.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/r6cNUjS8cm0/r6cNUjS8cm0_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/BJeWUs05KQ/BJeWUs05KQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/sCZbhBvqQaU/sCZbhBvqQaU_span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/sCZbhBvqQaU/sCZbhBvqQaU_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/nVofoXjTmA_/nVofoXjTmA__span.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/nVofoXjTmA_/nVofoXjTmA__origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/_WnwtieRHxM/_WnwtieRHxM_origin.pdf filter=lfs diff=lfs merge=lfs -text
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parse/train/dgtpE6gKjHn/dgtpE6gKjHn_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6731 |
+
parse/train/xHKVVHGDOEk/xHKVVHGDOEk_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6732 |
+
parse/train/xHKVVHGDOEk/xHKVVHGDOEk_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6733 |
+
parse/train/xHKVVHGDOEk/xHKVVHGDOEk_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6734 |
+
parse/train/wl0Kr_jqM2a/wl0Kr_jqM2a_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6735 |
+
parse/train/wl0Kr_jqM2a/wl0Kr_jqM2a_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6736 |
+
parse/train/wl0Kr_jqM2a/wl0Kr_jqM2a_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6737 |
+
parse/train/r1gelyrtwH/r1gelyrtwH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6738 |
+
parse/train/r1gelyrtwH/r1gelyrtwH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6739 |
+
parse/train/r1gelyrtwH/r1gelyrtwH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6740 |
+
parse/train/HygtHnR5tQ/HygtHnR5tQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6741 |
+
parse/train/HygtHnR5tQ/HygtHnR5tQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6742 |
+
parse/train/HygtHnR5tQ/HygtHnR5tQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6743 |
+
parse/train/ByJWeR1AW/ByJWeR1AW_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6744 |
+
parse/train/ByJWeR1AW/ByJWeR1AW_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6745 |
+
parse/train/ByJWeR1AW/ByJWeR1AW_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6746 |
+
parse/train/RYcgfqmAOHh/RYcgfqmAOHh_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6747 |
+
parse/train/RYcgfqmAOHh/RYcgfqmAOHh_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6748 |
+
parse/train/RYcgfqmAOHh/RYcgfqmAOHh_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6749 |
+
parse/train/Skltqh4KvB/Skltqh4KvB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6750 |
+
parse/train/Skltqh4KvB/Skltqh4KvB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6751 |
+
parse/train/Skltqh4KvB/Skltqh4KvB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6752 |
+
parse/train/HkgHk3RctX/HkgHk3RctX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6753 |
+
parse/train/HkgHk3RctX/HkgHk3RctX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6754 |
+
parse/train/HkgHk3RctX/HkgHk3RctX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6755 |
+
parse/train/WrotwUEJO59/WrotwUEJO59_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6756 |
+
parse/train/WrotwUEJO59/WrotwUEJO59_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6757 |
+
parse/train/WrotwUEJO59/WrotwUEJO59_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6758 |
+
parse/train/N9oPAFcuYWX/N9oPAFcuYWX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6759 |
+
parse/train/N9oPAFcuYWX/N9oPAFcuYWX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6760 |
+
parse/train/N9oPAFcuYWX/N9oPAFcuYWX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6761 |
+
parse/train/B1l08oAct7/B1l08oAct7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6762 |
+
parse/train/B1l08oAct7/B1l08oAct7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6763 |
+
parse/train/B1l08oAct7/B1l08oAct7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6764 |
+
parse/train/1-j4VLSHApJ/1-j4VLSHApJ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6765 |
+
parse/train/1-j4VLSHApJ/1-j4VLSHApJ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6766 |
+
parse/train/1-j4VLSHApJ/1-j4VLSHApJ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6767 |
+
parse/train/kHromd7SNA/kHromd7SNA_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6768 |
+
parse/train/kHromd7SNA/kHromd7SNA_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6769 |
+
parse/train/kHromd7SNA/kHromd7SNA_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6770 |
+
parse/train/S1xq3oR5tQ/S1xq3oR5tQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6771 |
+
parse/train/S1xq3oR5tQ/S1xq3oR5tQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6772 |
+
parse/train/S1xq3oR5tQ/S1xq3oR5tQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6773 |
+
parse/train/Syzn9i05Ym/Syzn9i05Ym_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6774 |
+
parse/train/Syzn9i05Ym/Syzn9i05Ym_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6775 |
+
parse/train/Syzn9i05Ym/Syzn9i05Ym_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6776 |
+
parse/train/kzPtpIpF8o/kzPtpIpF8o_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6777 |
+
parse/train/kzPtpIpF8o/kzPtpIpF8o_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6778 |
+
parse/train/kzPtpIpF8o/kzPtpIpF8o_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6779 |
+
parse/train/zGsRcuoR5-0/zGsRcuoR5-0_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6780 |
+
parse/train/zGsRcuoR5-0/zGsRcuoR5-0_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6781 |
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parse/train/zGsRcuoR5-0/zGsRcuoR5-0_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6782 |
+
parse/train/xCy9thPPTb_/xCy9thPPTb__origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6783 |
+
parse/train/xCy9thPPTb_/xCy9thPPTb__span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6784 |
+
parse/train/xCy9thPPTb_/xCy9thPPTb__layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6785 |
+
parse/train/HWX5j6Bv_ih/HWX5j6Bv_ih_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6786 |
+
parse/train/HWX5j6Bv_ih/HWX5j6Bv_ih_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6787 |
+
parse/train/HWX5j6Bv_ih/HWX5j6Bv_ih_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6788 |
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parse/train/HJe4Cp4KwH/HJe4Cp4KwH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6789 |
+
parse/train/HJe4Cp4KwH/HJe4Cp4KwH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6790 |
+
parse/train/HJe4Cp4KwH/HJe4Cp4KwH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6791 |
+
parse/train/SJx0q1rtvS/SJx0q1rtvS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6792 |
+
parse/train/SJx0q1rtvS/SJx0q1rtvS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6793 |
+
parse/train/SJx0q1rtvS/SJx0q1rtvS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6794 |
+
parse/train/RHY_9ZVcTa_/RHY_9ZVcTa__span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6795 |
+
parse/train/RHY_9ZVcTa_/RHY_9ZVcTa__layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6796 |
+
parse/train/RHY_9ZVcTa_/RHY_9ZVcTa__origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6797 |
+
parse/train/rJo9n9Feg/rJo9n9Feg_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6798 |
+
parse/train/rJo9n9Feg/rJo9n9Feg_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6799 |
+
parse/train/rJo9n9Feg/rJo9n9Feg_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6800 |
+
parse/train/HJxyZkBKDr/HJxyZkBKDr_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6801 |
+
parse/train/HJxyZkBKDr/HJxyZkBKDr_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6802 |
+
parse/train/HJxyZkBKDr/HJxyZkBKDr_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6803 |
+
parse/train/MNVjrDpu6Yo/MNVjrDpu6Yo_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6804 |
+
parse/train/MNVjrDpu6Yo/MNVjrDpu6Yo_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6805 |
+
parse/train/MNVjrDpu6Yo/MNVjrDpu6Yo_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6806 |
+
parse/train/SJxzPsAqFQ/SJxzPsAqFQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6807 |
+
parse/train/SJxzPsAqFQ/SJxzPsAqFQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6808 |
+
parse/train/SJxzPsAqFQ/SJxzPsAqFQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6809 |
+
parse/train/BJxH22EKPS/BJxH22EKPS_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6810 |
+
parse/train/BJxH22EKPS/BJxH22EKPS_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 6811 |
+
parse/train/BJxH22EKPS/BJxH22EKPS_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
parse/train/1-j4VLSHApJ/1-j4VLSHApJ_layout.pdf
ADDED
|
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|
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|
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parse/train/1-j4VLSHApJ/1-j4VLSHApJ_span.pdf
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parse/train/Aw96fN64soV/Aw96fN64soV.md
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|
| 1 |
+
# Neural Analysis and Synthesis: Reconstructing Speech from Self-Supervised Representations
|
| 2 |
+
|
| 3 |
+
Hyeong-Seok Choi1,4 Juheon Lee1,4 Wansoo Kim1,4
|
| 4 |
+
|
| 5 |
+
Jie Hwan Lee4 Hoon Heo4 Kyogu Lee1,2,3,4
|
| 6 |
+
|
| 7 |
+
1MARG, Department of Intelligence and Information, Seoul National University 2GSAI 3AIIS 4Supertone Inc.
|
| 8 |
+
|
| 9 |
+
{kekepa15, juheon2, wansookim, kglee}@snu.ac.kr, {wiswisbus, hoon}@supertone.ai
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# Abstract
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We present a neural analysis and synthesis (NANSY) framework that can manipulate voice, pitch, and speed of an arbitrary speech signal. Most of the previous works have focused on using information bottleneck to disentangle analysis features for controllable synthesis, which usually results in poor reconstruction quality. We address this issue by proposing a novel training strategy based on information perturbation. The idea is to perturb information in the original input signal (e.g., formant, pitch, and frequency response), thereby letting synthesis networks selectively take essential attributes to reconstruct the input signal. Because NANSY does not need any bottleneck structures, it enjoys both high reconstruction quality and controllability. Furthermore, NANSY does not require any labels associated with speech data such as text and speaker information, but rather uses a new set of analysis features, i.e., wav2vec feature and newly proposed pitch feature, Yingram, which allows for fully self-supervised training. Taking advantage of fully selfsupervised training, NANSY can be easily extended to a multilingual setting by simply training it with a multilingual dataset. The experiments show that NANSY can achieve significant improvement in performance in several applications such as zero-shot voice conversion, pitch shift, and time-scale modification 1.
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# 1 Introduction
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Analyzing and synthesizing an arbitrary speech signal is inarguably a significant research topic that has been studied for decades. Traditionally, this has been studied in the digital signal processing (DSP) field using fundamental methods such as sinusoidal modeling or linear predictive coding (LPC), and it is the analysis and synthesis framework that lies at the heart of those fundamental methods $[ \sqrt { 2 6 } , \bigstar ]$ . These traditional methods, however, are limited in terms of controllability because the decomposed representations are still low-level representations. It is obvious that the closer we decompose a signal into high-level/interpretable representations, the more we gain access to the controllability. Given this consideration, we aim to design a neural analysis and synthesis (NANSY) framework by decomposing a speech signal into analysis features that represent pronunciation, timbre, pitch, and energy. The decomposed representations can be manipulated and re-synthesized, enabling users to manipulate speech signals in various ways.
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It is worth noting that many similar ideas have been recently proposed in the context of voice conversion applications. We categorize the previous works in two ways, i.e., 1. Text-based approach,
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2. Information bottleneck approach. The first approach exploits the fact that the text modality is inherently disentangled from the speaker identity. One of the most popular text-based approaches is to use a pre-trained automatic speech recognition (ASR) network to extract a phonetic posteriogram (PPG) and use it as a linguistic feature [41]. Then, combining the PPG with the target speaker information, the features are re-synthesized to a speech signal. Another alternative approach is to directly use text scripts by aligning it to a paired source signal $\mathbb { \lVert 3 3 \rVert }$ . Although these ideas have shown promising results, it is important to note that these approaches have common problems. First, in order to extract the PPG features, it is required to train an ASR network in a supervised manner, which demands a lot of paired text and waveform datasets. Additionally, the language dependency of the ASR network limits the model’s capability to be extended to multilingual settings or languages with low-resources. To address these concerns, efforts have been made to divert from using the text information and the most popular approach is to use an information bottleneck. The key idea is to restrict the information flow by reducing time/channel dimension, and normalizing/quantizing intermediate representations [36, 10, 51]. Although these ideas have been explored in many ways, one critical problem is that there exists an inevitable trade-off between the degree of disentanglement and the reconstruction quality. In other words, there is a trade-off between speaker similarity and the preservation of original content such as linguistic and pitch information.
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To avoid the major concern of the text-based approach, we suggest to use two analysis features which are wav2vec and a newly proposed feature, Yingram. In order to preserve the linguistic information without any text information, we utilize wav2vec $2 . 0 \mathbb { H }$ , trained on 53 languages in total $\mathbb { m }$ . While the features from wav2vec 2.0 have mostly been used for a downstream task, we seek the possibility of using them for an upstream/generation task. In addition, we propose a new feature that can effectively represent and control pitch information. Although it is the fundamental frequency $( f _ { 0 } )$ that is mostly used to represent the pitch information, $f _ { 0 }$ is sometimes ill-defined when there exists sub-harmonics in the signal (e.g., vocal fry) [16, 1, 2]. We address this issue by proposing a controllable but more abstract feature than $f _ { 0 }$ that still includes information such as sub-harmonics. Because the proposed feature is heavily inspired by the famous Yin algorithm $\mathbb { \lVert \lambda \rVert }$ , we refer to this feature as Yingram.
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Although the analysis features above have enough information to reconstruct the original speech signal, we have found that the information in the proposed analysis features share common information such as pitch and timbre. To disentangle the common information so each feature can control a specific attribute for its desired purpose (e.g., wav2vec linguistic information only, Yingram pitch information only), we propose an information perturbation approach, a simple yet effective solution to this problem. The idea is to simply perturb all the information we do not want to control from the input features, thereby training the neural network to not extract the undesirable attributes from the features. Through this way, the model no longer suffers from the unavoidable trade-off between reconstruction quality and feature disentanglement, unlike the information bottleneck approach.
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Lastly, we would like to deal with unseen languages at test time. To this end, we propose a new test-time self-adaptation (TSA) strategy. The proposed self-adaptation strategy does not fine-tune the model parameters but only the input linguistic feature, which consequently modifies the mispronounced parts of the reconstructed sample. Because the proposed TSA requires only a single sample at test-time, it adds a large flexibility for the model to be used in many scenarios (e.g., low-resource language).
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The contributions of this paper are as follows:
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• We propose a neural analysis and synthesis (NANSY) framework that can be trained in a fully self-supervised manner (no text, no speaker information needed). The proposed method is based on a new set of analysis features and information perturbation.
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• The proposed model can be used for various applications, including zero-shot voice conversion, formant preserving pitch shift, and time-scale modification.
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• We propose a new test-time self-adaptation (TSA) technique than can be used even on unseen languages using only a single test-time speech sample.
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Figure 1: The overview of the training procedure and information flow of the proposed neural analysis and synthesis (NANSY) framework. The waveform is first perturbed using functions $f$ and $g$ . $f$ perturbs formant, pitch, and frequency response. $g$ perturbs formant and frequency response while preserving pitch. w2v denotes wav2vec encoder and spk denotes a speaker embedding network. L, P, S, E denotes Linguistic, Pitch, Speaker, and Energy information, respectively. The tilde symbol is attached when the information is perturbed using the perturbation functions. The dashed boxes denote the modules that are being trained.
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# 2.1 Analysis Features
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Linguistic To reconstruct an intelligible speech signal, it is crucial to extract rich linguistic information from the speech signal. To this end, we resort to XLSR-53: a wav2vec 2.0 model pre-trained on $5 6 \mathrm { k }$ hours of speech in 53 languages [11]. The extracted features from XLSR-53 have shown superior performance on downstream tasks such as ASR, especially on low-resource languages. We conjecture, therefore, that the extracted features from this model can provide language-agnostic linguistic information. Now the question is, from which layer should the features be extracted? Recently, it has been reported that the representation from different layers of wav2vec 2.0 exhibit different characteristics. Especially, Shah et al. $\mathbb { \lVert 3 9 \rVert }$ showed that it is the output from the middle layer that has the most relevant characteristics to pronunciation2. In light of this empirical observation, we decided to use the intermediate features of XLSR-53. More specifically, we used the output from the 12th layer of the 24-layer transformer encoder.
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Speaker Perhaps the most common approach to extract speaker embeddings is to first train a speaker recognition network in a supervised manner and then reuse the network for the generation task, assuming that the speaker embedding from the trained network can represent the characteristics of unseen speakers [21, 36]. Here we would like to take one step further and assume that we do not have speaker labels to train a speaker recognition network in a supervised manner. To mitigate this disadvantage, we again use the representation from XLSR-53, which makes the proposed method fully self-supervised. To determine which layer of XLSR-53 to extract the representation from, we first analyzed the features from each layer. Specifically, we averaged the representation of each layer along the time-axis and visualized utterances of 20 randomly selected speakers from the VCTK dataset using TSNE [47, 46]. In Fig. $\bigstar$ we can observe that the representation from the 1st layer of XLSR-53 already forms clusters for each speaker, while the latter layers (especially the last layer) tend to lack them. Note that this is in accordance with the previous observation in $\dot { \mathbb { I B } }$ . Taking this into consideration, we train a speaker embedding network that uses the 1st layer of XLSR-53 as an input. For the speaker embedding network, we borrow the neural architecture from a state-of-the-art speaker recognition network $\pmb { \mathbb { I } } \pmb { \ 4 } \|$ , which is based on 1D-convolutional neural networks (1D-CNN) with an attentive statistics pooling layer. The speaker embedding was $L _ { 2 }$ -normalized before conditioning. The speaker embeddings of seen and unseen speakers during training are also shown in Fig. 2.
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Pitch Due to the irregular periodicity of the glottal pulse, we often hear creaky voice in speech, which is usually manifested as jitter or sub-harmonics in signals. This makes hard for $f _ { 0 }$ trackers to estimate $f _ { 0 }$ because the $f _ { 0 }$ itself is not well defined in such cases [16, 1, 2]. We take a hint from the popular Yin algorithm to address this issue. The Yin algorithm uses the cumulative mean normalized difference function $d _ { t } ^ { \prime } ( \tau )$ to extract frame-wise features from a raw waveform, which is defined as follows,
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Figure 2: The visualization of intermediate representations of XLSR-53 using TSNE.
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$$
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d _ { t } ^ { \prime } ( \tau ) = \left\{ \begin{array} { l l } { 1 , } & { \mathrm { i f } \ \tau = 0 } \\ { d _ { t } ( \tau ) / \sum _ { j = 1 } ^ { \tau } d _ { t } ( j ) , } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
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$$
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The $d _ { t } ( \tau )$ is a difference function that outputs a small value when there exists a periodicity on time-lag $\tau$ and it is defined as follows,
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$$
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d _ { t } ( \tau ) = \sum _ { j = 1 } ^ { W } \left( x _ { j } - x _ { j + \tau } \right) ^ { 2 } = r _ { t } ( 0 ) + r _ { t + \tau } ( 0 ) - 2 r _ { t } ( \tau ) ,
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$$
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where $t , \tau , W$ , and $r _ { t }$ denote frame index, time lag, window size, and the auto-correlation function, respectively. After some post processing steps, the Yin algorithm selects $f _ { 0 }$ from multiple $f _ { 0 }$ candidates. See $\mathbb { \lVert \lambda \rVert }$ for more details. Rather than explicitly selecting $f _ { 0 }$ , we would like to train the network to generate pitch harmonics from the output of the function $d _ { t } ^ { \prime } ( \tau )$ . However, $d _ { t } ^ { \prime } ( \tau )$ itself is limited to be used as a pitch feature because it lacks controllability, unlike $f _ { 0 }$ . Therefore, we propose Yingram $Y$ by converting the time-lag axis to the midi-scale axis as follows,
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$$
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Y _ { t } ( m ) = \frac { d _ { t } ^ { \prime } ( \lceil c ( m ) \rceil ) - d _ { t } ^ { \prime } ( \lfloor c ( m ) \rfloor ) } { \lceil c ( m ) \rceil - \lfloor c ( m ) \rfloor } \cdot ( c ( m ) - \lfloor c ( m ) \rfloor ) + d _ { t } ^ { \prime } ( \lfloor c ( m ) \rfloor ) ,
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$$
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$$
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c ( m ) = \frac { s r } { 4 4 0 \cdot 2 ^ { ( \frac { m - 6 9 } { 1 2 } ) } } ,
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$$
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where $m$ , $c ( m )$ , and $s r$ denote midi note, midi-to-lag conversion function, and sampling rate, respectively. We set 20 bins of Yingram to represent a semitone range. In addition, we set Yingram to represent the frequency between $1 0 . 7 7 \mathrm { h z }$ and $1 0 0 0 . 4 0 \mathrm { h z }$ by setting $W$ to 2048 and the range of $\tau$ between 22 and 2047. In the training stage, the input to the synthesis network is the frequency range between $2 5 . 1 1 \mathrm { h z }$ and $4 3 0 . 1 9 \mathrm { h z }$ , which is shown as scope in Fig. $\textcircled { 3 }$ After the training is finished, we can change the pitch by shifting the scope. That is, in the inference stage, one could simply change the pitch of the speech signal by shifting the scope. For example, if we move the scope down 20 bins, the pitch can be raised by a semitone.
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Figure 3: The visualization of Yingram and the corresponding mel spectrogram.
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Energy For the energy feature, we simply took an average from a log-mel spectrogram along the frequency axis.
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# 2.2 Synthesis Network
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Figure 4: The outputs of $\mathcal { G } _ { S }$ and $\mathcal { G } _ { F }$ . The two outputs from each generator are summed to reconstruct a mel spectrogram.
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It is well-known that speech production can be explained by source-filter theory. Inspired by this, we separate synthesis networks into two parts, source generator $\mathcal { G } _ { S }$ and filter generator $\mathcal { G } _ { F }$ . While the energy and speaker features are common inputs for both generators, $\mathcal { G } _ { S }$ and $\mathcal { G } _ { F }$ differ in that they take Yingram and wav2vec features, respectively. Because the acoustic feature can be interpreted as a sum of source and filter in the log magnitude domain, we incorporate inductive bias in the model by summing the outputs from each generator similarly to $[ [ 2 5 ] ]$ . As will be discussed in more detail in the next section, even though the training loss is only defined using mel spectrograms, the network learns to separately generate the spectral envelope and pitch harmonics as shown in Fig. $4 .$ Note that this separation not only provides the interpretability to the model but also enables formant preserving pitch shifting. To summarize, the acoustic feature, mel spectrogram $\hat { M }$ , is generated as follows,
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$$
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\hat { M } = \mathcal { G } _ { S } ( \mathrm { Y i n g r a m } , S , E ) + \mathcal { G } _ { F } ( \mathrm { w a v } 2 \mathrm { v e c } , S , E ) ,
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$$
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where $S$ and $E$ denote speaker embedding and energy features. We used stacks of 1D-CNN layers with gated linear units (GLU) $\pmb { \mathbb { I } }$ for generators. The detailed neural architecture of the generator is described in Appendix $\mathbf { B } .$ Note that each generator shares the same neural architecture. The only difference is the input features to the networks. Finally, the generated mel spectrogram is converted to waveform using the pre-trained HiFi-GAN vocoder $\pmb { \Vert 2 4 \Vert }$ .
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# 3 Training
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# 3.1 Information Perturbation
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In our initial experiments, a neural network can be easily trained to reconstruct mel spectrograms using only the wav2vec feature. This implies that the wav2vec feature contains not only rich linguistic information but also information related to pitch and speaker. For that reason, we would like to train $\mathcal { G } _ { F }$ to selectively extract only the linguistic-related information from the wav2vec feature, not pitch and speaker information. In addition, we would like to train $\mathcal { G } _ { S }$ to selectively extract only the pitch-related information from the Yingram feature, not speaker information. To this end, we propose to perturb the information included in input waveform $x$ by using three functions that are 1. formant shifting $( f s )$ , 2. pitch randomization $( p r )$ , and 3. random frequency shaping using a parametric equalizer $( p e q ) \ L ^ { | 3 | }$ We applied a function $f$ on the wav2vec input, which is a chain of all three functions as follows, $\bar { f ( x ) } = \bar { f s ( p r ( p e q ( x ) ) ) }$ . On the Yingram side, we applied function $g$ , which is a chain of two functions $f s$ and peq so that $f _ { 0 }$ information is still preserved as follows, $g ( x ) = f s ( p e q ( x ) )$ . This way, we expect $\mathcal { G } _ { F }$ to take only the linguistic-related information from the wav2vec feature, and $\mathcal { G } _ { S }$ to take only pitch-related from the Yingram feature. Since the wav2vec and Yingram features can no longer provide the speaker-related information, the control of speaker information becomes uniquely dependent on the speaker embedding. The overview of the information flow is shown in Fig.
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1. The hyperparameters of the perturbation functions are described more in Appendix A.
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# 3.2 Training Loss
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We used L1 loss between the generated mel spectrogram $\hat { M }$ and ground truth mel spectrogram $M$ to train the generators and speaker embedding network. However, it is well-known that the speech synthesis networks trained with L1 or L2 loss suffer from over-smootheness of the generated acoustic feature, which results in poor quality of the speech signal. Therefore, in addition to the L1 loss, we used the recent speaker conditional generative adversarial training method to mitigate this issue [8]. Writing the discriminator as $\bar { \cal D } ( { \cal M } , { \pmb { c } } _ { + } , { \pmb { c } } _ { - } ) : = \sigma ( h ( { \cal M } , { \pmb { c } } _ { + } , { \pmb { c } } _ { - } ) )$ , Choi et al. $\checkmark$ proposed to use projection conditioning $\mathbb { \left[ \left[ 2 \right] \right] }$ not only with the positive pairs but also with the negative pairs as follows,
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$$
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h ( M , \pmb { c } _ { + } , \pmb { c } _ { - } ) = \psi ( \phi ( M ) ) + \pmb { c } _ { + } ^ { T } \phi ( M ) - \pmb { c } _ { - } ^ { T } \phi ( M ) ,
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$$
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where $M$ denotes a mel spectrogram, $\sigma ( \cdot )$ denotes a sigmoid function, $c _ { + }$ denotes a speaker embedding from a positively paired input speech sample, and $c _ { - }$ denotes a speaker embedding from a randomly sampled speech utterance. $\phi ( \cdot )$ denotes an output from the intermediate layer of discriminator and $\psi ( \cdot )$ denotes a function that maps input vector to a scalar value. The detailed neural architecture of $\mathcal { D }$ is shown in Appendix B. The loss functions for discriminator $L _ { \mathcal { D } }$ and generator $L _ { \mathcal { G } }$ are as follows:
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$$
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\begin{array} { r l } & { L _ { \mathcal { D } } = - \mathbb { E } _ { ( M , c _ { + } , c _ { - } ) \sim p _ { d a t a } , \hat { M } \sim p _ { g e n } } [ l o g ( \sigma ( h ( M , c _ { + } , \pmb { c } _ { - } ) ) ) - l o g ( \sigma ( h ( \hat { M } , \pmb { c } _ { + } , \pmb { c } _ { - } ) ) ) ] , } \\ & { L _ { \mathcal { G } } = - \mathbb { E } _ { ( M , c _ { + } , c _ { - } ) \sim p _ { d a t a } , \hat { M } \sim p _ { g e n } } [ l o g ( \sigma ( h ( \hat { M } , \pmb { c } _ { + } , \pmb { c } _ { - } ) ) ) ] + | M - \hat { M } | . } \end{array}
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$$
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# 3.3 Test-time Self-Adaptation
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Although the synthesis network can reconstruct an intelligible speech from the wav2vec feature in most cases, we observed that the network sometimes outputs speech signals with wrong pronunciation, especially when tested on unseen languages. To alleviate this problem, we propose to modify only the input representation, that is, the wav2vec feature, without having to train the whole network again from scratch. As shown in Fig. $\textcircled { 5 }$ we first compute L1 loss between the generated mel spectrogram $\hat { M }$ and ground truth mel spectrogram $M$ in the test-time. Then, we
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Figure 5: The illustration of TSA.
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update only the parameterized wav2vec feature using the backpropagation signal from the loss. Note that the loss gradient (shown in red) is backpropagated only through the filter generator. Because this test-time training scheme requires only a single test-time sample and updates the input parameters by targeting the test-time sample itself, we call it test-time self-adaptation (TSA).
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# 4 Experiments
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# 4.1 Implementation Details
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Dataset To train NANSY on English, we used two datasets, i.e., 1. VCTK4 [47], 2. train-clean-360 subset of LibriTTS3 [54]. We trained the model using $90 \%$ of samples for each speaker. The speakers of train-clean-360 were included to the training set only when the total length of speech samples exceeds 15 minutes. To test on English speech samples we used two datasets; 1. For the seen speaker test we used $10 \%$ unseen utterances of VCTK. 2. For the unseen speaker test we used test-clean subset of LibriTTS.
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To train NANSY on multi-language, we used $\mathrm { C S S 1 0 ^ { 3 } }$ dataset $\lVert \overline { { 3 2 } } \rVert$ . CSS10 includes 10 speakers and each speaker use different language. Note that there is no English speaking speaker included in CSS10. To train the model, we used $90 \%$ of samples for each speaker. To test on multilingual speech samples, we used the rest $10 \%$ unseen utterances of CSS10.
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Training We used $2 2 , 0 5 0 \mathrm { h z }$ sampling rate for every analysis feature except for wav2vec input that takes waveform with the sampling rate of $1 6 { , } 0 0 0 \mathrm { h z }$ . We used 80 bands for mel spectrogram, where FFT, window, and hop size were set to 1024, 1024, and 256, respectively. The samples were randomly cropped approximately to 1.47-second, which results in 128 mel spectrogram frames. The networks were trained using Adam optimizer with $\beta _ { 1 } = 0 . 5$ and $\beta _ { 2 } = 0 . 9$ . The learning rate was fixed to $1 0 ^ { - 4 }$ . We trained every model using one RTX 3090 with batch size 32. The training was done after 50 epochs.
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# 4.2 Reconstruction
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For the reconstruction (analysis and synthesis) tests, we report character error rate (CER $( \% )$ ), and 5-scale mean opinon score (MOS ([1-5])), 5-scale degradation mean opinion score (DMOS ([1-5])). For MOS, higher is better. For DMOS and CER, lower is better. To estimate the characters from speech samples, we used google cloud ASR API. For MOS and DMOS, we used amazon mechanical turk (MTurk). The details of MOS and DMOS are shown in Appendix D.
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Yingram vs $f _ { 0 }$ We compared two models trained with Yingram and $f _ { 0 }$ to check which pitch feature shows more robust reconstruction performance. We used RAPT algorithm for $f _ { 0 }$ estimation $\lVert \overline { { 4 2 } } \rVert$ which is known as a reliable $f _ { 0 }$ tracker among many other algorithms $\overline { { \| 2 2 } }$ . Because RAPT algorithm works sufficiently well in most cases, we first manually listened to the reconstructed samples using the model trained with $f _ { 0 }$ . We first chose 30 reconstructed samples in the testset that failed to faithfully reconstruct the original samples using the model trained with $f _ { 0 }$ . After that we reconstructed the same 30 samples using the model trained with Yingram. Finally, we conducted ABX test to ask participants which of the two samples (A and B) sounds closer to the original sample (X). The ABX test was conducted on MTurk. The results showed that the participants chose Yingram with a chance of $6 8 . 3 \%$ . This shows that Yingram can be used as a more robust pitch feature than $f _ { 0 }$ , when $f _ { 0 }$ cannot be accurately estimated.
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Reconstruction test We randomly sampled 50 speech samples from VCTK (seen speaker) and sampled another 50 speech samples from test-clean subset of LibriTTS (unseen speaker) to test the reconstruction performance of NANSY trained on English datasets. The results in Table $\bigstar$ shows that NANSY can perform high quality analysis and synthesis task. In addition, to test if the proposed framework can cover various languages, we trained and tested it with the multilingual dataset, CSS10. We randomly sampled 100 speech samples from CSS10 to test the reconstruction performance of NANSY trained on multi-language. The results are shown in Table $\boxed { 2 }$ Although we do not impose any explicit labels for each language, the model was able to reconstruct various languages with high quality. The results of CER on each language is shown in Fig. $\bigtriangledown$ MUL.
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Table 1: English reconstruction results.
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<table><tr><td></td><td>CER</td><td>MOS</td><td>DMOS</td></tr><tr><td>GT</td><td>n/a</td><td>4.28 ± 0.09</td><td>n/a</td></tr><tr><td>Recon</td><td>5.6</td><td>4.18 ± 0.09</td><td>1.93 ± 0.09</td></tr></table>
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Table 2: Multilingual reconstruction results.
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<table><tr><td></td><td>CER</td><td>MOS</td><td>DMOS</td></tr><tr><td>GT</td><td>n/a</td><td>4.19 ± 0.08</td><td>n/a</td></tr><tr><td>Recon</td><td>7.3</td><td>4.14 ± 0.09</td><td>1.74 ± 0.09</td></tr></table>
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Test-time self-adaptation To test the proposed test-time self-adaptation (TSA), we compared the CER performance of NANSY trained in three different configurations, 1. English (ENG), 2. English with TSA (ENG-TSA), 3. Multi-language (MUL). For every experiment, we iteratively updated the wav2vec feature 100 times. The results are shown in Fig. $\triangledown$ Naturally, MUL generally showed better CER performance than other configurations. Interestingly, however, ENG-TSA sometimes showed similar or even better performance than MUL, which shows the effectiveness of the proposed TSA technique.
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Figure 6: The CER results on 10 languages (NL: Dutch, HU: Hungarian, FR: French, JP: Japanese, CH: Chinese, RU: Russian, FI: Finnish, GR: Greek, ES: Spanish, DE: German).
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# 4.3 Voice conversion
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NANSY can perform zero-shot voice conversion by simply passing the desired target speech utterance to the speaker embedding network. We first compared the English voice conversion performance of NANSY with recently proposed zero-shot voice conversion models. Next, we tested the multilingual voice conversion performance. Finally, we tested unseen language voice conversion for both seen speaker and unseen speaker targets. Note that in every voice conversion experiment, we shifted the median pitch of a source utterance to the median pitch of a target utterance by shifting the scope of Yingram. We measured naturalness with 5-scale mean opinion score (MOS [1-5]). Speaker similarity (SSIM $( \% )$ ) were measured with a binary decision and uncertainty options, following $\pmb { \Vert 5 0 \Vert }$ . For MOS and SSIM, we again used MTurk. The details of MOS and SSIM are shown in Appendix D. One of the crucial criteria for evaluating the quality of the converted samples is to check the intelligibility of them. Previous zero-shot voice conversion models, however, have only reported MOS or SSIM and have been negligent on assessing the intelligibility of the converted samples [36, 10, 51]. To this end, we report character error rate (CER $( \% ) _ { , }$ ) between estimated characters of source and converted pairs. To estimate the characters from speech samples, we used google cloud ASR API.
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Algorithm comparison Here, we report three source-to-target speaker conversion settings, 1. seen-to-seen (many-to-many, M2M), 2. unseen-to-seen (any-to-many, A2M), 3. unseen-to-unseen (any-to-any, A2A). For every setting, we considered 4 gender-to-gender combination, i.e., male-tomale $( { \mathrm { m } } 2 { \mathrm { m } } )$ , male-to-female (m2f), female-to-male $( \mathrm { f } 2 \mathrm { m } )$ , and female-to-female (f2f). For M2M setting, we randomly sampled 25 seen speakers from VCTK and randomly assigned 2 random speakers from VCTK, resulting in 200 $( = 2 5 { \times } 2 { \times } 4 )$ conversion pairs in total. For A2M setting, we randomly sampled 10 seen speakers from test-clean subset of LibriTTS and randomly assigned 2 random speakers from VCTK resulting in 80 $( = 1 0 \times 2 \times 4 )$ ) conversion pairs in total. For A2A setting, we randomly sampled 10 seen speakers from test-clean subset of LibriTTS and randomly assigned 2 random speakers from test-clean subset of LibriTTS resulting in 80 $( = 1 0 \times 2 \times 4 )$ conversion pairs in total. We trained three baseline models with official implementations - ${ \mathrm { V Q V C } } +$ [51], AdaIN $[ \mathbb { 1 0 } ]$ , AUTOVC $\pmb { \mathbb { B } } 6 \|$ - using the same dataset and mel spectrogram configuration as NANSY. For a fair comparison, we used a pre-trained HiFi-GAN vocoder for every model. Table $\triangledown$ shows that NANSY significantly outperforms previous models in terms of every evaluation measure. This implies that the proposed information perturbation approach does not suffer from the trade-off between CER and SSIM unlike information bottleneck approaches. The SSIM results for all possible gender-to-gender combinations are shown in Fig. 9 in Appendix C.
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<table><tr><td>一</td><td></td><td>M2M</td><td></td><td></td><td>A2M</td><td></td><td></td><td>A2A</td><td></td></tr><tr><td></td><td></td><td>[CER[%] MOS[1-5]</td><td></td><td></td><td>]SSIM[%]|CER[%]MOS[1-5]</td><td></td><td></td><td>SSIM[%]|CER[%] MOS[1-5]</td><td>SSIM[%]</td></tr><tr><td>SRC as TGT</td><td>n/a</td><td>4.23 ± 0.05</td><td>0</td><td>n/a</td><td>4.28 ± 0.09</td><td>0.60</td><td>n/a</td><td>4.26 ± 0.07</td><td>0.25</td></tr><tr><td>TGT as TGT</td><td>n/a</td><td>4.32 ± 0.05</td><td>94.9</td><td>n/a</td><td>4.29 ± 0.05</td><td>92.4</td><td>n/a</td><td>4.27 ± 0.07</td><td>96.2</td></tr><tr><td>VQVC+</td><td>54.0</td><td>1.76 ± 0.05</td><td>54.5</td><td>74.7</td><td>1.73 ± 0.11</td><td>15.6</td><td>69.3</td><td>1.83 ± 0.09</td><td>13.8</td></tr><tr><td>AdaIN</td><td>62.9</td><td>2.22 ± 0.07</td><td>24.0</td><td>79.6</td><td>1.92 ± 0.12</td><td>18.1</td><td>59.3</td><td>2.12 ± 0.10</td><td>21.2</td></tr><tr><td>AUTOVC</td><td>31.7</td><td>3.41 ±0.06</td><td>47.3</td><td>36.1</td><td>2.74 ±0.11</td><td>33.2</td><td>28.2</td><td>2.59 ±0.08</td><td>23.3</td></tr><tr><td>NANSY</td><td>7.5</td><td>3.79 ± 0.07</td><td>91.4</td><td>7.6</td><td>3.73 ± 0.05</td><td>88.1</td><td>8.6</td><td>3.44 ± 0.07</td><td>64.6</td></tr></table>
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Table 3: Evaluation results on English voice conversion. SRC and TGT denote, source and target, respectively.
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Multilingual voice conversion We tested multilingual voice conversion performance with the model trained on the multilingual dataset, CSS10. We randomly sampled 50 samples for each language speaker from CSS10 and assigned random single target speaker from CSS10 for each source language, resulting in 500 $( = 5 0 \times 1 0 )$ ) conversion pairs in total. The results in Table 4 show that the proposed framework can successfully perform multilingual voice conversion by training it with the multilingual dataset. However, there is still a room for improvement for multilingual voice conversion when comparing to the results in Table 3, where NANSY is just trained on English.
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Unseen language voice conversion We tested the voice conversion performance on unseen source languages using NANSY trained on English. We tested the performance on two settings, 1. unseen source language (CSS10) to seen target voice (VCTK) and 2. unseen source language (CSS10) to unseen target voice (CSS10). For the first experiment, we randomly sampled 50 samples for each unseen language speaker from CSS10 and assigned random English target speakers from VCTK, resulting in 500 $( = 5 0 \times 1 0 )$ conversion pairs in total. The second experiment was conducted identically to the multilingual voice conversion experiment setting. The results in Table $\boxed { 5 }$ shows that NANSY can be successfully extended even to unseen language sources, although there was a decrease on SSIM compared to the results in Table 4 $( 6 9 . 5 \% \bar { } 6 \bar { 1 } . 0 \% )$ ).
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Table 4: The multilingual voice conversion results. The model was trained using only CSS10.
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<table><tr><td></td><td>|CER MOS</td><td>SSIM</td></tr><tr><td></td><td>TGT as TGT|n/a 4.23 ±0.06</td><td>98.0</td></tr><tr><td>NANSY</td><td>18.8 3.68 ±0.09</td><td>69.5</td></tr></table>
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<table><tr><td></td><td colspan="3">Seen Speaker</td></tr><tr><td></td><td>|CER MOS</td><td>SSIM|CER</td><td>MOS SSIM</td></tr><tr><td>TGT as TGT NANSY</td><td>n/a 4.24 ±0.07 14.8 3.75 ± 0.10</td><td>100 90.0</td><td>n/a 4.23 ± 0.06 92.0 15.6 3.76 ± 0.09 61.0</td></tr></table>
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Table 5: The voice conversion results on unseen language dataset, CSS10. The model was trained using only the English datasets.
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# 4.4 Pitch shift and time-scale modification
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To check the robustness of pitch shift (PS) and time-scale modification (TSM) performance of NANSY, we compared it with other robust algorithms, i.e., PSOLA 5 and WORLD vocoder [29, 28].
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Pitch shift We tested PS performance with 5 semitone ranges, -6, -3, 0, 3, 6. The pitch was changed by shifting the scope of the proposed Yingram feature. Note that $\mathbf { \bar { \rho } } _ { 0 } ,$ was used to check analysissynthesis performance. We randomly selected 20 samples for each semitone range from $10 \%$ unseen utterances of VCTK. We evaluated the naturalness of speech samples with MOS on MTurk. The results in Table $6$ show that NANSY generally outperforms algorithms such as PSOLA and WORLD vocoder on PS task.
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Time-scale modification We tested TSM performance with 5 time-scale ratios, 1/2, 1/1.5, 1, 1.5, 2. The time-scale was modified by simply manipulating the hop length of the analysis features. Note that $\cdot _ { 1 } \cdot$ was used to check analysis-synthesis performance. We randomly selected 20 samples for each time-scale ratio from $10 \%$ unseen utterances of VCTK. We evaluated the naturalness of speech samples with MOS on MTurk. The results in Table 7 show that NANSY achieved competitive performance on TSM compared to the well-established PSOLA and WORLD vocoder.
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<table><tr><td></td><td>-6</td><td>-3</td><td>0</td><td>3</td><td>6</td></tr><tr><td>WORLD 28</td><td>3.433.53</td><td></td><td></td><td>33.853.63</td><td>3.53</td></tr><tr><td>PSOLA 四</td><td></td><td></td><td></td><td></td><td>3.633.553.933.753.48</td></tr><tr><td>NANSY</td><td>3.60 3.68</td><td></td><td>4.05</td><td>3.90</td><td>3.78</td></tr></table>
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Table 6: Pitch shift results.
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Table 7: Time-scale modification results.
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<table><tr><td></td><td>1/2</td><td>1/1.5 1</td><td>1.5</td><td>2</td></tr><tr><td>WORLD I28I</td><td>2.403.603.833.38</td><td></td><td></td><td>83.03</td></tr><tr><td>PSOLA 四</td><td>2.55 3.66 3.953.68</td><td></td><td></td><td>2.85</td></tr><tr><td>NANSY</td><td>2.453.63</td><td>33.983.70</td><td></td><td>2.93</td></tr></table>
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# 5 Related Works
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Self-supervised representation learning and synthesis of speech There has been an increasing interest in the self-supervised learning methods within the machine learning and speech processing community. Oord et al. [31] first proposed to use noise contrastive estimation loss to train speech representations. Baevski et al. [4] extended this idea by integrating masked language modeling $\mathbb { \lVert 1 5 \rVert }$ . Another popular self-supervised learning method for speech representation is to train a neural network by targeting multiple self-supervision tasks [34, 37]. Most recently, $\pmb { \Vert 3 5 \Vert }$ used the discrete disentangled self-supervised representations to re-synthesize them into a waveform. Although using the discrete units has its own advantage in that it is disentangled with speaker information, we found that an inaccurate quantization process often leads to mispronounced samples, which is why we turned to use continuous representation as it can provide more accurate results on linguistic information.
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Zero-shot voice conversion Research on zero-shot voice conversion has been most actively conducted through the information bottleneck approach. Qian et al. $\textcircled { \lvert 3 6 \rvert }$ proposed to perform zero-shot voice conversion by utilizing the pre-trained speaker recognition network and information bottleneck by carefully designing the bottleneck of an auto-encoder. Inspired by the success of style conversion in computer vision, Chou and Lee $\mathbb { \ m }$ also focused on restricting the information flow using instance normalization $\pm \ddagger { 4 }$ and adaptive instance normalization $\mathbb { \lVert 1 9 \rVert }$ . Lastly, Wu et al. [51] used multiple vector quantization layers $\bar { \mathbb { B } } \bar { 0 } \bar { 1 }$ to restrict the information flow.
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Inductive bias for audio generation It has been shown that neural networks can be combined with traditional speech/sound production models for efficient and strong performance. One of the speech production models that has been integrated with neural networks is the source-filter model. By modeling source and filter components with deep networks, it has been used for applications such as vocoder [49, 23, 45] and acoustic feature generation $\lVert 2 5 \rVert$ . Furthermore, Engel et al. $\mathbb { \ m }$ proposed to integrate a harmonic plus noise model $\bar { \big \| } \bar { 3 8 } \bar { \big \| }$ and neural networks to produce natural audio signal.
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Consistency learning Learning representations by augmenting the data has been one of the key ideas to leverage the performance of classification tasks [43, 52]. This shares the similar idea with the proposed information perturbation strategy in that the data is perturbed so that the neural network must learn to ignore the perturbations and learn the consistency from the data. However, the key difference between the consistency learning and the information perturbation is that the information perturbation method is designed for “generative” task and that it is the “decoder” (e.g., Generator) that is trained to selectively take the essential attributes to reconstruct the signal from the given perturbed representations.
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# 6 Conclusions and Discussion
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In this work, we proposed a neural analysis and synthesis framework (NANSY) that can perform zero-shot voice conversion, formant preserving pitch shift, and time-scale modification with a single model. The proposed model can be trained in a fully self-supervised manner, that is, it can be trained without any labeled data such as text or speaker information. We showed the effectiveness of the proposed information perturbation approach by showing the voice conversion results on various settings. Furthermore, we showed the effectiveness of the proposed TSA method by testing it on unseen languages, which shows the possibility of NANSY to be extended on low-resource languages. Although the proposed method empowers controllability over several attributes of a speech signal, it is still limited in terms of lacking controllability over linguistic information. As a future work, therefore, we would like to investigate on a hybrid approach that integrates text information as a side input so that the user can manipulate even the linguistic information in the speech signal. Finally, to prevent the proposed framework being used maliciously (e.g., voice phishing), it would be important to develop a detection algorithm that can discriminate a synthesized speech sample from a real speech sample. To examine the potential of such a detection system, we have tried using the trained Discriminator from the NANSY framework, which is expected to discriminate fake samples from real samples. We measured the accuracy of classifying 185 reconstructed samples and 185 ground truth samples. In addition, we measured the accuracy of classifying 560 voice conversion samples and 560 ground truth samples. The accuracy was $9 1 . 4 \%$ on the reconstruction set and $9 5 . 5 \%$ on the voice conversion set. This shows the possibility of Discriminator being used as a byproduct network to discriminate real speech samples from fake speech samples. However, we also found that Discriminator is prone to being deceived by the generated samples from other speech generative models as Discriminator was not jointly trained with those generative models. Therefore, we expect more robust synthesized speech detection algorithms to be developed in the future such as [48, 40, 9, 6].
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# Broader Impacts
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The proposed NANSY framework shows that a generative model can benefit from self-supervised representations. By choosing proper domain specific “information perturbation” functions, we believe that one can achieve controllable generative modeling in a fully self-supervised way. The information perturbation training strategy may also be used for other modalities and facilitate self-supervised representation learning methods too. With the proposed training framework, one can manipulate various aspects of speech samples. Among the various controllabilities, it is rather obvious that the voice conversion technique can be misused and potentially harm other people. More concretely, there are possible scenarios where it is being used by random unidentified users and contributing to spreading fake news. In addition, it can raise concerns about biometric security systems based on speech. To mitigate such issues, the proposed system should not be released without a consent so that it cannot be easily used by random users with malicious intentions. That being said, there is still a potential for this technology to be used by unidentified users. As a more solid solution, therefore, we believe a detection system that can discriminate between fake and real speech should be developed. The preliminary results of the detection system is reported in section 6.
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# Acknowledgments and Disclosure of Funding
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This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) [NO.2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University)]
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# Checklist
|
| 301 |
+
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| 302 |
+
1. For all authors...
|
| 303 |
+
|
| 304 |
+
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We described the goal and contribution of this paper and conducted experiments accordingly.
|
| 305 |
+
(b) Did you describe the limitations of your work? [Yes] We explained the limitation of this work in the conclusion.
|
| 306 |
+
(c) Did you discuss any potential negative societal impacts of your work? [Yes] We discussed the potential negative societal impacts (e.g., voice phishing) of our work in the conclusion.
|
| 307 |
+
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have read the ethics review guidelines.
|
| 308 |
+
|
| 309 |
+
2. If you are including theoretical results...
|
| 310 |
+
|
| 311 |
+
(a) Did you state the full set of assumptions of all theoretical results? [No] We do not include theoretical results.
|
| 312 |
+
(b) Did you include complete proofs of all theoretical results? [No] We do not include theoretical results.
|
| 313 |
+
|
| 314 |
+
3. If you ran experiments...
|
| 315 |
+
|
| 316 |
+
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] The code is proprietary.
|
| 317 |
+
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See section $4 . 1$ and Appendix A.
|
| 318 |
+
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We have included the error bars for crowdsourcing evaluation.
|
| 319 |
+
|
| 320 |
+
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See section 4.1.
|
| 321 |
+
|
| 322 |
+
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
|
| 323 |
+
|
| 324 |
+
(a) If your work uses existing assets, did you cite the creators? [Yes] We cited all three datasets we used for experiments.
|
| 325 |
+
(b) Did you mention the license of the assets? [Yes] See section 4.1.
|
| 326 |
+
(c) Did you include any new assets either in the supplemental material or as a URL? [No] We did not curate/release any new assets.
|
| 327 |
+
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We cited the paper of the datasets we are using in which they explain all the details regarding speaker recruitment.
|
| 328 |
+
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] We used speaker labels to split the datasets as described in section 4.1.
|
| 329 |
+
|
| 330 |
+
5. If you used crowdsourcing or conducted research with human subjects...
|
| 331 |
+
|
| 332 |
+
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [Yes] We have attached the screenshots of MTurk instructions in Appendix D.
|
| 333 |
+
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [Yes] This work is approved by IRB (IRB No. 2105/004-008). We have announced that the de-identified information such as worker ID will be collected through MTurk instructions.
|
| 334 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [Yes] It is shown in Appendix D.
|
parse/train/Aw96fN64soV/Aw96fN64soV_content_list.json
ADDED
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| 1 |
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[
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| 2 |
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{
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| 3 |
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"type": "text",
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| 4 |
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"text": "Neural Analysis and Synthesis: Reconstructing Speech from Self-Supervised Representations ",
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| 5 |
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"text_level": 1,
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},
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{
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"type": "text",
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"text": "Hyeong-Seok Choi1,4 Juheon Lee1,4 Wansoo Kim1,4 ",
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| 17 |
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{
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"type": "text",
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"text": "Jie Hwan Lee4 Hoon Heo4 Kyogu Lee1,2,3,4 ",
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| 28 |
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"type": "text",
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"text": "1MARG, Department of Intelligence and Information, Seoul National University 2GSAI 3AIIS 4Supertone Inc. ",
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"type": "text",
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"text": "{kekepa15, juheon2, wansookim, kglee}@snu.ac.kr, {wiswisbus, hoon}@supertone.ai ",
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"type": "text",
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"text": "Abstract ",
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| 61 |
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"text_level": 1,
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"type": "text",
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| 72 |
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"text": "We present a neural analysis and synthesis (NANSY) framework that can manipulate voice, pitch, and speed of an arbitrary speech signal. Most of the previous works have focused on using information bottleneck to disentangle analysis features for controllable synthesis, which usually results in poor reconstruction quality. We address this issue by proposing a novel training strategy based on information perturbation. The idea is to perturb information in the original input signal (e.g., formant, pitch, and frequency response), thereby letting synthesis networks selectively take essential attributes to reconstruct the input signal. Because NANSY does not need any bottleneck structures, it enjoys both high reconstruction quality and controllability. Furthermore, NANSY does not require any labels associated with speech data such as text and speaker information, but rather uses a new set of analysis features, i.e., wav2vec feature and newly proposed pitch feature, Yingram, which allows for fully self-supervised training. Taking advantage of fully selfsupervised training, NANSY can be easily extended to a multilingual setting by simply training it with a multilingual dataset. The experiments show that NANSY can achieve significant improvement in performance in several applications such as zero-shot voice conversion, pitch shift, and time-scale modification 1. ",
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"type": "text",
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"text": "1 Introduction ",
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| 84 |
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"text_level": 1,
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| 85 |
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"type": "text",
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"text": "Analyzing and synthesizing an arbitrary speech signal is inarguably a significant research topic that has been studied for decades. Traditionally, this has been studied in the digital signal processing (DSP) field using fundamental methods such as sinusoidal modeling or linear predictive coding (LPC), and it is the analysis and synthesis framework that lies at the heart of those fundamental methods $[ \\sqrt { 2 6 } , \\bigstar ]$ . These traditional methods, however, are limited in terms of controllability because the decomposed representations are still low-level representations. It is obvious that the closer we decompose a signal into high-level/interpretable representations, the more we gain access to the controllability. Given this consideration, we aim to design a neural analysis and synthesis (NANSY) framework by decomposing a speech signal into analysis features that represent pronunciation, timbre, pitch, and energy. The decomposed representations can be manipulated and re-synthesized, enabling users to manipulate speech signals in various ways. ",
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"text": "It is worth noting that many similar ideas have been recently proposed in the context of voice conversion applications. We categorize the previous works in two ways, i.e., 1. Text-based approach, ",
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"type": "text",
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"text": "2. Information bottleneck approach. The first approach exploits the fact that the text modality is inherently disentangled from the speaker identity. One of the most popular text-based approaches is to use a pre-trained automatic speech recognition (ASR) network to extract a phonetic posteriogram (PPG) and use it as a linguistic feature [41]. Then, combining the PPG with the target speaker information, the features are re-synthesized to a speech signal. Another alternative approach is to directly use text scripts by aligning it to a paired source signal $\\mathbb { \\lVert 3 3 \\rVert }$ . Although these ideas have shown promising results, it is important to note that these approaches have common problems. First, in order to extract the PPG features, it is required to train an ASR network in a supervised manner, which demands a lot of paired text and waveform datasets. Additionally, the language dependency of the ASR network limits the model’s capability to be extended to multilingual settings or languages with low-resources. To address these concerns, efforts have been made to divert from using the text information and the most popular approach is to use an information bottleneck. The key idea is to restrict the information flow by reducing time/channel dimension, and normalizing/quantizing intermediate representations [36, 10, 51]. Although these ideas have been explored in many ways, one critical problem is that there exists an inevitable trade-off between the degree of disentanglement and the reconstruction quality. In other words, there is a trade-off between speaker similarity and the preservation of original content such as linguistic and pitch information. ",
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"text": "To avoid the major concern of the text-based approach, we suggest to use two analysis features which are wav2vec and a newly proposed feature, Yingram. In order to preserve the linguistic information without any text information, we utilize wav2vec $2 . 0 \\mathbb { H }$ , trained on 53 languages in total $\\mathbb { m }$ . While the features from wav2vec 2.0 have mostly been used for a downstream task, we seek the possibility of using them for an upstream/generation task. In addition, we propose a new feature that can effectively represent and control pitch information. Although it is the fundamental frequency $( f _ { 0 } )$ that is mostly used to represent the pitch information, $f _ { 0 }$ is sometimes ill-defined when there exists sub-harmonics in the signal (e.g., vocal fry) [16, 1, 2]. We address this issue by proposing a controllable but more abstract feature than $f _ { 0 }$ that still includes information such as sub-harmonics. Because the proposed feature is heavily inspired by the famous Yin algorithm $\\mathbb { \\lVert \\lambda \\rVert }$ , we refer to this feature as Yingram. ",
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"type": "text",
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"text": "Although the analysis features above have enough information to reconstruct the original speech signal, we have found that the information in the proposed analysis features share common information such as pitch and timbre. To disentangle the common information so each feature can control a specific attribute for its desired purpose (e.g., wav2vec linguistic information only, Yingram pitch information only), we propose an information perturbation approach, a simple yet effective solution to this problem. The idea is to simply perturb all the information we do not want to control from the input features, thereby training the neural network to not extract the undesirable attributes from the features. Through this way, the model no longer suffers from the unavoidable trade-off between reconstruction quality and feature disentanglement, unlike the information bottleneck approach. ",
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| 140 |
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"text": "Lastly, we would like to deal with unseen languages at test time. To this end, we propose a new test-time self-adaptation (TSA) strategy. The proposed self-adaptation strategy does not fine-tune the model parameters but only the input linguistic feature, which consequently modifies the mispronounced parts of the reconstructed sample. Because the proposed TSA requires only a single sample at test-time, it adds a large flexibility for the model to be used in many scenarios (e.g., low-resource language). ",
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"type": "text",
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"text": "The contributions of this paper are as follows: ",
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| 162 |
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"type": "text",
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"text": "• We propose a neural analysis and synthesis (NANSY) framework that can be trained in a fully self-supervised manner (no text, no speaker information needed). The proposed method is based on a new set of analysis features and information perturbation. \n• The proposed model can be used for various applications, including zero-shot voice conversion, formant preserving pitch shift, and time-scale modification. \n• We propose a new test-time self-adaptation (TSA) technique than can be used even on unseen languages using only a single test-time speech sample. ",
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"type": "image",
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"img_path": "images/df31aea3e54082daf0a7a221cea97a36043eb99b754577ea6cef9306de10aa7a.jpg",
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"image_caption": [
|
| 185 |
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"Figure 1: The overview of the training procedure and information flow of the proposed neural analysis and synthesis (NANSY) framework. The waveform is first perturbed using functions $f$ and $g$ . $f$ perturbs formant, pitch, and frequency response. $g$ perturbs formant and frequency response while preserving pitch. w2v denotes wav2vec encoder and spk denotes a speaker embedding network. L, P, S, E denotes Linguistic, Pitch, Speaker, and Energy information, respectively. The tilde symbol is attached when the information is perturbed using the perturbation functions. The dashed boxes denote the modules that are being trained. "
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| 187 |
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"type": "text",
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| 198 |
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"text": "2.1 Analysis Features ",
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| 199 |
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"text_level": 1,
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"type": "text",
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"text": "Linguistic To reconstruct an intelligible speech signal, it is crucial to extract rich linguistic information from the speech signal. To this end, we resort to XLSR-53: a wav2vec 2.0 model pre-trained on $5 6 \\mathrm { k }$ hours of speech in 53 languages [11]. The extracted features from XLSR-53 have shown superior performance on downstream tasks such as ASR, especially on low-resource languages. We conjecture, therefore, that the extracted features from this model can provide language-agnostic linguistic information. Now the question is, from which layer should the features be extracted? Recently, it has been reported that the representation from different layers of wav2vec 2.0 exhibit different characteristics. Especially, Shah et al. $\\mathbb { \\lVert 3 9 \\rVert }$ showed that it is the output from the middle layer that has the most relevant characteristics to pronunciation2. In light of this empirical observation, we decided to use the intermediate features of XLSR-53. More specifically, we used the output from the 12th layer of the 24-layer transformer encoder. ",
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"type": "text",
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"text": "Speaker Perhaps the most common approach to extract speaker embeddings is to first train a speaker recognition network in a supervised manner and then reuse the network for the generation task, assuming that the speaker embedding from the trained network can represent the characteristics of unseen speakers [21, 36]. Here we would like to take one step further and assume that we do not have speaker labels to train a speaker recognition network in a supervised manner. To mitigate this disadvantage, we again use the representation from XLSR-53, which makes the proposed method fully self-supervised. To determine which layer of XLSR-53 to extract the representation from, we first analyzed the features from each layer. Specifically, we averaged the representation of each layer along the time-axis and visualized utterances of 20 randomly selected speakers from the VCTK dataset using TSNE [47, 46]. In Fig. $\\bigstar$ we can observe that the representation from the 1st layer of XLSR-53 already forms clusters for each speaker, while the latter layers (especially the last layer) tend to lack them. Note that this is in accordance with the previous observation in $\\dot { \\mathbb { I B } }$ . Taking this into consideration, we train a speaker embedding network that uses the 1st layer of XLSR-53 as an input. For the speaker embedding network, we borrow the neural architecture from a state-of-the-art speaker recognition network $\\pmb { \\mathbb { I } } \\pmb { \\ 4 } \\|$ , which is based on 1D-convolutional neural networks (1D-CNN) with an attentive statistics pooling layer. The speaker embedding was $L _ { 2 }$ -normalized before conditioning. The speaker embeddings of seen and unseen speakers during training are also shown in Fig. 2. ",
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"type": "text",
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| 232 |
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"text": "Pitch Due to the irregular periodicity of the glottal pulse, we often hear creaky voice in speech, which is usually manifested as jitter or sub-harmonics in signals. This makes hard for $f _ { 0 }$ trackers to estimate $f _ { 0 }$ because the $f _ { 0 }$ itself is not well defined in such cases [16, 1, 2]. We take a hint from the popular Yin algorithm to address this issue. The Yin algorithm uses the cumulative mean normalized difference function $d _ { t } ^ { \\prime } ( \\tau )$ to extract frame-wise features from a raw waveform, which is defined as follows, ",
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"type": "image",
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"img_path": "images/7ea07596546947f4e230d1d518e51138b57a0f4403db7be53b5ad549ebf49ca9.jpg",
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| 244 |
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"image_caption": [
|
| 245 |
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"Figure 2: The visualization of intermediate representations of XLSR-53 using TSNE. "
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| 246 |
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],
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| 247 |
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"type": "equation",
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"img_path": "images/d620de40baf2fd1ff1bc1bf7bc79934fe7f34073169fe89b3ea4e0662e177a30.jpg",
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"text": "$$\nd _ { t } ^ { \\prime } ( \\tau ) = \\left\\{ \\begin{array} { l l } { 1 , } & { \\mathrm { i f } \\ \\tau = 0 } \\\\ { d _ { t } ( \\tau ) / \\sum _ { j = 1 } ^ { \\tau } d _ { t } ( j ) , } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.\n$$",
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"text_format": "latex",
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"type": "text",
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"text": "The $d _ { t } ( \\tau )$ is a difference function that outputs a small value when there exists a periodicity on time-lag $\\tau$ and it is defined as follows, ",
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| 287 |
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362
|
| 288 |
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|
| 289 |
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"page_idx": 3
|
| 290 |
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|
| 291 |
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{
|
| 292 |
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"type": "equation",
|
| 293 |
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"img_path": "images/1ce01bd02b794370b95fedfa1d27fd63a5a52cf63101a1ddbfc374941b08fc3f.jpg",
|
| 294 |
+
"text": "$$\nd _ { t } ( \\tau ) = \\sum _ { j = 1 } ^ { W } \\left( x _ { j } - x _ { j + \\tau } \\right) ^ { 2 } = r _ { t } ( 0 ) + r _ { t + \\tau } ( 0 ) - 2 r _ { t } ( \\tau ) ,\n$$",
|
| 295 |
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"text_format": "latex",
|
| 296 |
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"bbox": [
|
| 297 |
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312,
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| 298 |
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| 299 |
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| 300 |
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| 301 |
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|
| 302 |
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| 303 |
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|
| 304 |
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{
|
| 305 |
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"type": "text",
|
| 306 |
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"text": "where $t , \\tau , W$ , and $r _ { t }$ denote frame index, time lag, window size, and the auto-correlation function, respectively. After some post processing steps, the Yin algorithm selects $f _ { 0 }$ from multiple $f _ { 0 }$ candidates. See $\\mathbb { \\lVert \\lambda \\rVert }$ for more details. Rather than explicitly selecting $f _ { 0 }$ , we would like to train the network to generate pitch harmonics from the output of the function $d _ { t } ^ { \\prime } ( \\tau )$ . However, $d _ { t } ^ { \\prime } ( \\tau )$ itself is limited to be used as a pitch feature because it lacks controllability, unlike $f _ { 0 }$ . Therefore, we propose Yingram $Y$ by converting the time-lag axis to the midi-scale axis as follows, ",
|
| 307 |
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| 315 |
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| 316 |
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"type": "equation",
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| 317 |
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"img_path": "images/425ae697925d6d6445e649fbb7ec514d0abb5ca661691b14f32b24d981be2f0c.jpg",
|
| 318 |
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"text": "$$\nY _ { t } ( m ) = \\frac { d _ { t } ^ { \\prime } ( \\lceil c ( m ) \\rceil ) - d _ { t } ^ { \\prime } ( \\lfloor c ( m ) \\rfloor ) } { \\lceil c ( m ) \\rceil - \\lfloor c ( m ) \\rfloor } \\cdot ( c ( m ) - \\lfloor c ( m ) \\rfloor ) + d _ { t } ^ { \\prime } ( \\lfloor c ( m ) \\rfloor ) ,\n$$",
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| 319 |
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| 320 |
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| 324 |
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| 325 |
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| 326 |
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| 327 |
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},
|
| 328 |
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{
|
| 329 |
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"type": "equation",
|
| 330 |
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"img_path": "images/3f47929d32ba73b1d134e5841565c85b34563b65ffd032dfe74379fd1830ac08.jpg",
|
| 331 |
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"text": "$$\nc ( m ) = \\frac { s r } { 4 4 0 \\cdot 2 ^ { ( \\frac { m - 6 9 } { 1 2 } ) } } ,\n$$",
|
| 332 |
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"text_format": "latex",
|
| 333 |
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"bbox": [
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| 335 |
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| 336 |
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| 337 |
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| 338 |
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| 341 |
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| 342 |
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"type": "text",
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| 343 |
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"text": "where $m$ , $c ( m )$ , and $s r$ denote midi note, midi-to-lag conversion function, and sampling rate, respectively. We set 20 bins of Yingram to represent a semitone range. In addition, we set Yingram to represent the frequency between $1 0 . 7 7 \\mathrm { h z }$ and $1 0 0 0 . 4 0 \\mathrm { h z }$ by setting $W$ to 2048 and the range of $\\tau$ between 22 and 2047. In the training stage, the input to the synthesis network is the frequency range between $2 5 . 1 1 \\mathrm { h z }$ and $4 3 0 . 1 9 \\mathrm { h z }$ , which is shown as scope in Fig. $\\textcircled { 3 }$ After the training is finished, we can change the pitch by shifting the scope. That is, in the inference stage, one could simply change the pitch of the speech signal by shifting the scope. For example, if we move the scope down 20 bins, the pitch can be raised by a semitone. ",
|
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| 352 |
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| 353 |
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"type": "image",
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| 354 |
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"img_path": "images/2dc9bfcfd416ccc2c8e65c9a4b700caf125a911da552a301b7255af070136526.jpg",
|
| 355 |
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"image_caption": [
|
| 356 |
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"Figure 3: The visualization of Yingram and the corresponding mel spectrogram. "
|
| 357 |
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],
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| 358 |
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| 368 |
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"type": "text",
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| 369 |
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"text": "Energy For the energy feature, we simply took an average from a log-mel spectrogram along the frequency axis. ",
|
| 370 |
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"type": "text",
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| 380 |
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"text": "2.2 Synthesis Network ",
|
| 381 |
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"text_level": 1,
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| 391 |
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"type": "image",
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| 392 |
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"img_path": "images/8e73c09f29547f84328621c584d5700bf5b642b371093c7d55519169b8e892ff.jpg",
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| 393 |
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"image_caption": [
|
| 394 |
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"Figure 4: The outputs of $\\mathcal { G } _ { S }$ and $\\mathcal { G } _ { F }$ . The two outputs from each generator are summed to reconstruct a mel spectrogram. "
|
| 395 |
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],
|
| 396 |
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|
| 397 |
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| 406 |
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"type": "text",
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| 407 |
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"text": "It is well-known that speech production can be explained by source-filter theory. Inspired by this, we separate synthesis networks into two parts, source generator $\\mathcal { G } _ { S }$ and filter generator $\\mathcal { G } _ { F }$ . While the energy and speaker features are common inputs for both generators, $\\mathcal { G } _ { S }$ and $\\mathcal { G } _ { F }$ differ in that they take Yingram and wav2vec features, respectively. Because the acoustic feature can be interpreted as a sum of source and filter in the log magnitude domain, we incorporate inductive bias in the model by summing the outputs from each generator similarly to $[ [ 2 5 ] ]$ . As will be discussed in more detail in the next section, even though the training loss is only defined using mel spectrograms, the network learns to separately generate the spectral envelope and pitch harmonics as shown in Fig. $4 .$ Note that this separation not only provides the interpretability to the model but also enables formant preserving pitch shifting. To summarize, the acoustic feature, mel spectrogram $\\hat { M }$ , is generated as follows, ",
|
| 408 |
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| 415 |
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| 416 |
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| 417 |
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"type": "equation",
|
| 418 |
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"img_path": "images/baffdda6183a3fd5ca0967fca3939a5da01a44c8fba0bfe4700748862ad662c3.jpg",
|
| 419 |
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"text": "$$\n\\hat { M } = \\mathcal { G } _ { S } ( \\mathrm { Y i n g r a m } , S , E ) + \\mathcal { G } _ { F } ( \\mathrm { w a v } 2 \\mathrm { v e c } , S , E ) ,\n$$",
|
| 420 |
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"text_format": "latex",
|
| 421 |
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"bbox": [
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| 422 |
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| 423 |
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| 428 |
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| 429 |
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| 430 |
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"type": "text",
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| 431 |
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"text": "where $S$ and $E$ denote speaker embedding and energy features. We used stacks of 1D-CNN layers with gated linear units (GLU) $\\pmb { \\mathbb { I } }$ for generators. The detailed neural architecture of the generator is described in Appendix $\\mathbf { B } .$ Note that each generator shares the same neural architecture. The only difference is the input features to the networks. Finally, the generated mel spectrogram is converted to waveform using the pre-trained HiFi-GAN vocoder $\\pmb { \\Vert 2 4 \\Vert }$ . ",
|
| 432 |
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| 439 |
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| 440 |
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| 441 |
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"type": "text",
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| 442 |
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"text": "3 Training ",
|
| 443 |
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"text_level": 1,
|
| 444 |
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| 451 |
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| 452 |
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{
|
| 453 |
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"type": "text",
|
| 454 |
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"text": "3.1 Information Perturbation ",
|
| 455 |
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"text_level": 1,
|
| 456 |
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"bbox": [
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"page_idx": 4
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|
| 464 |
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{
|
| 465 |
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"type": "text",
|
| 466 |
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"text": "In our initial experiments, a neural network can be easily trained to reconstruct mel spectrograms using only the wav2vec feature. This implies that the wav2vec feature contains not only rich linguistic information but also information related to pitch and speaker. For that reason, we would like to train $\\mathcal { G } _ { F }$ to selectively extract only the linguistic-related information from the wav2vec feature, not pitch and speaker information. In addition, we would like to train $\\mathcal { G } _ { S }$ to selectively extract only the pitch-related information from the Yingram feature, not speaker information. To this end, we propose to perturb the information included in input waveform $x$ by using three functions that are 1. formant shifting $( f s )$ , 2. pitch randomization $( p r )$ , and 3. random frequency shaping using a parametric equalizer $( p e q ) \\ L ^ { | 3 | }$ We applied a function $f$ on the wav2vec input, which is a chain of all three functions as follows, $\\bar { f ( x ) } = \\bar { f s ( p r ( p e q ( x ) ) ) }$ . On the Yingram side, we applied function $g$ , which is a chain of two functions $f s$ and peq so that $f _ { 0 }$ information is still preserved as follows, $g ( x ) = f s ( p e q ( x ) )$ . This way, we expect $\\mathcal { G } _ { F }$ to take only the linguistic-related information from the wav2vec feature, and $\\mathcal { G } _ { S }$ to take only pitch-related from the Yingram feature. Since the wav2vec and Yingram features can no longer provide the speaker-related information, the control of speaker information becomes uniquely dependent on the speaker embedding. The overview of the information flow is shown in Fig. ",
|
| 467 |
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|
| 475 |
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|
| 476 |
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"type": "text",
|
| 477 |
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"text": "1. The hyperparameters of the perturbation functions are described more in Appendix A. ",
|
| 478 |
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| 486 |
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|
| 487 |
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"type": "text",
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| 488 |
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"text": "3.2 Training Loss ",
|
| 489 |
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"text_level": 1,
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| 499 |
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"type": "text",
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| 500 |
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"text": "We used L1 loss between the generated mel spectrogram $\\hat { M }$ and ground truth mel spectrogram $M$ to train the generators and speaker embedding network. However, it is well-known that the speech synthesis networks trained with L1 or L2 loss suffer from over-smootheness of the generated acoustic feature, which results in poor quality of the speech signal. Therefore, in addition to the L1 loss, we used the recent speaker conditional generative adversarial training method to mitigate this issue [8]. Writing the discriminator as $\\bar { \\cal D } ( { \\cal M } , { \\pmb { c } } _ { + } , { \\pmb { c } } _ { - } ) : = \\sigma ( h ( { \\cal M } , { \\pmb { c } } _ { + } , { \\pmb { c } } _ { - } ) )$ , Choi et al. $\\checkmark$ proposed to use projection conditioning $\\mathbb { \\left[ \\left[ 2 \\right] \\right] }$ not only with the positive pairs but also with the negative pairs as follows, ",
|
| 501 |
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|
| 510 |
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"type": "text",
|
| 511 |
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"text": "",
|
| 512 |
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| 521 |
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"type": "equation",
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| 522 |
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"img_path": "images/b194bff851e39fdd44dc91fd6d19ffb23841385fd610c45e029ea9f027b73d9c.jpg",
|
| 523 |
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"text": "$$\nh ( M , \\pmb { c } _ { + } , \\pmb { c } _ { - } ) = \\psi ( \\phi ( M ) ) + \\pmb { c } _ { + } ^ { T } \\phi ( M ) - \\pmb { c } _ { - } ^ { T } \\phi ( M ) ,\n$$",
|
| 524 |
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"text_format": "latex",
|
| 525 |
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"bbox": [
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|
| 533 |
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{
|
| 534 |
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"type": "text",
|
| 535 |
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"text": "where $M$ denotes a mel spectrogram, $\\sigma ( \\cdot )$ denotes a sigmoid function, $c _ { + }$ denotes a speaker embedding from a positively paired input speech sample, and $c _ { - }$ denotes a speaker embedding from a randomly sampled speech utterance. $\\phi ( \\cdot )$ denotes an output from the intermediate layer of discriminator and $\\psi ( \\cdot )$ denotes a function that maps input vector to a scalar value. The detailed neural architecture of $\\mathcal { D }$ is shown in Appendix B. The loss functions for discriminator $L _ { \\mathcal { D } }$ and generator $L _ { \\mathcal { G } }$ are as follows: ",
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"type": "equation",
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"img_path": "images/e02f68cdc9e8347bf452e60dfc698538c88c2d65dbb646c67ff73495c84f2b4f.jpg",
|
| 547 |
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"text": "$$\n\\begin{array} { r l } & { L _ { \\mathcal { D } } = - \\mathbb { E } _ { ( M , c _ { + } , c _ { - } ) \\sim p _ { d a t a } , \\hat { M } \\sim p _ { g e n } } [ l o g ( \\sigma ( h ( M , c _ { + } , \\pmb { c } _ { - } ) ) ) - l o g ( \\sigma ( h ( \\hat { M } , \\pmb { c } _ { + } , \\pmb { c } _ { - } ) ) ) ] , } \\\\ & { L _ { \\mathcal { G } } = - \\mathbb { E } _ { ( M , c _ { + } , c _ { - } ) \\sim p _ { d a t a } , \\hat { M } \\sim p _ { g e n } } [ l o g ( \\sigma ( h ( \\hat { M } , \\pmb { c } _ { + } , \\pmb { c } _ { - } ) ) ) ] + | M - \\hat { M } | . } \\end{array}\n$$",
|
| 548 |
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"text_format": "latex",
|
| 549 |
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| 557 |
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|
| 558 |
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"type": "text",
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| 559 |
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"text": "3.3 Test-time Self-Adaptation ",
|
| 560 |
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"text_level": 1,
|
| 561 |
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| 570 |
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"type": "text",
|
| 571 |
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"text": "Although the synthesis network can reconstruct an intelligible speech from the wav2vec feature in most cases, we observed that the network sometimes outputs speech signals with wrong pronunciation, especially when tested on unseen languages. To alleviate this problem, we propose to modify only the input representation, that is, the wav2vec feature, without having to train the whole network again from scratch. As shown in Fig. $\\textcircled { 5 }$ we first compute L1 loss between the generated mel spectrogram $\\hat { M }$ and ground truth mel spectrogram $M$ in the test-time. Then, we ",
|
| 572 |
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| 579 |
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},
|
| 580 |
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{
|
| 581 |
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"type": "image",
|
| 582 |
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"img_path": "images/c9b15cf4c0966b89bd0470d4e0592c780e88896a94b5d3e49b8fdd9cd169706c.jpg",
|
| 583 |
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"image_caption": [
|
| 584 |
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"Figure 5: The illustration of TSA. "
|
| 585 |
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],
|
| 586 |
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"image_footnote": [],
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| 587 |
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| 594 |
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},
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| 595 |
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{
|
| 596 |
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"type": "text",
|
| 597 |
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"text": "update only the parameterized wav2vec feature using the backpropagation signal from the loss. Note that the loss gradient (shown in red) is backpropagated only through the filter generator. Because this test-time training scheme requires only a single test-time sample and updates the input parameters by targeting the test-time sample itself, we call it test-time self-adaptation (TSA). ",
|
| 598 |
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{
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"type": "text",
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| 608 |
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"text": "4 Experiments ",
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| 609 |
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"type": "text",
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"text": "4.1 Implementation Details ",
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"text": "Dataset To train NANSY on English, we used two datasets, i.e., 1. VCTK4 [47], 2. train-clean-360 subset of LibriTTS3 [54]. We trained the model using $90 \\%$ of samples for each speaker. The speakers of train-clean-360 were included to the training set only when the total length of speech samples exceeds 15 minutes. To test on English speech samples we used two datasets; 1. For the seen speaker test we used $10 \\%$ unseen utterances of VCTK. 2. For the unseen speaker test we used test-clean subset of LibriTTS. ",
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"text": "To train NANSY on multi-language, we used $\\mathrm { C S S 1 0 ^ { 3 } }$ dataset $\\lVert \\overline { { 3 2 } } \\rVert$ . CSS10 includes 10 speakers and each speaker use different language. Note that there is no English speaking speaker included in CSS10. To train the model, we used $90 \\%$ of samples for each speaker. To test on multilingual speech samples, we used the rest $10 \\%$ unseen utterances of CSS10. ",
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"text": "Training We used $2 2 , 0 5 0 \\mathrm { h z }$ sampling rate for every analysis feature except for wav2vec input that takes waveform with the sampling rate of $1 6 { , } 0 0 0 \\mathrm { h z }$ . We used 80 bands for mel spectrogram, where FFT, window, and hop size were set to 1024, 1024, and 256, respectively. The samples were randomly cropped approximately to 1.47-second, which results in 128 mel spectrogram frames. The networks were trained using Adam optimizer with $\\beta _ { 1 } = 0 . 5$ and $\\beta _ { 2 } = 0 . 9$ . The learning rate was fixed to $1 0 ^ { - 4 }$ . We trained every model using one RTX 3090 with batch size 32. The training was done after 50 epochs. ",
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"type": "text",
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"text": "4.2 Reconstruction ",
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"text_level": 1,
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| 667 |
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"type": "text",
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"text": "For the reconstruction (analysis and synthesis) tests, we report character error rate (CER $( \\% )$ ), and 5-scale mean opinon score (MOS ([1-5])), 5-scale degradation mean opinion score (DMOS ([1-5])). For MOS, higher is better. For DMOS and CER, lower is better. To estimate the characters from speech samples, we used google cloud ASR API. For MOS and DMOS, we used amazon mechanical turk (MTurk). The details of MOS and DMOS are shown in Appendix D. ",
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"type": "text",
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"text": "Yingram vs $f _ { 0 }$ We compared two models trained with Yingram and $f _ { 0 }$ to check which pitch feature shows more robust reconstruction performance. We used RAPT algorithm for $f _ { 0 }$ estimation $\\lVert \\overline { { 4 2 } } \\rVert$ which is known as a reliable $f _ { 0 }$ tracker among many other algorithms $\\overline { { \\| 2 2 } }$ . Because RAPT algorithm works sufficiently well in most cases, we first manually listened to the reconstructed samples using the model trained with $f _ { 0 }$ . We first chose 30 reconstructed samples in the testset that failed to faithfully reconstruct the original samples using the model trained with $f _ { 0 }$ . After that we reconstructed the same 30 samples using the model trained with Yingram. Finally, we conducted ABX test to ask participants which of the two samples (A and B) sounds closer to the original sample (X). The ABX test was conducted on MTurk. The results showed that the participants chose Yingram with a chance of $6 8 . 3 \\%$ . This shows that Yingram can be used as a more robust pitch feature than $f _ { 0 }$ , when $f _ { 0 }$ cannot be accurately estimated. ",
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"type": "text",
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"text": "Reconstruction test We randomly sampled 50 speech samples from VCTK (seen speaker) and sampled another 50 speech samples from test-clean subset of LibriTTS (unseen speaker) to test the reconstruction performance of NANSY trained on English datasets. The results in Table $\\bigstar$ shows that NANSY can perform high quality analysis and synthesis task. In addition, to test if the proposed framework can cover various languages, we trained and tested it with the multilingual dataset, CSS10. We randomly sampled 100 speech samples from CSS10 to test the reconstruction performance of NANSY trained on multi-language. The results are shown in Table $\\boxed { 2 }$ Although we do not impose any explicit labels for each language, the model was able to reconstruct various languages with high quality. The results of CER on each language is shown in Fig. $\\bigtriangledown$ MUL. ",
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"type": "table",
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"img_path": "images/588525922963d00f9d901594fac2484d2d161b5afcbf99e49ee5dd8097f260c8.jpg",
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"table_caption": [
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| 712 |
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"Table 1: English reconstruction results. "
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| 713 |
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],
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"table_footnote": [],
|
| 715 |
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"table_body": "<table><tr><td></td><td>CER</td><td>MOS</td><td>DMOS</td></tr><tr><td>GT</td><td>n/a</td><td>4.28 ± 0.09</td><td>n/a</td></tr><tr><td>Recon</td><td>5.6</td><td>4.18 ± 0.09</td><td>1.93 ± 0.09</td></tr></table>",
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"type": "table",
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"img_path": "images/df7ad959716e840a1e4123c79a94ce91d8c2861021cfd6e9a71e513868a19b29.jpg",
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| 727 |
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"table_caption": [
|
| 728 |
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"Table 2: Multilingual reconstruction results. "
|
| 729 |
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],
|
| 730 |
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"table_footnote": [],
|
| 731 |
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"table_body": "<table><tr><td></td><td>CER</td><td>MOS</td><td>DMOS</td></tr><tr><td>GT</td><td>n/a</td><td>4.19 ± 0.08</td><td>n/a</td></tr><tr><td>Recon</td><td>7.3</td><td>4.14 ± 0.09</td><td>1.74 ± 0.09</td></tr></table>",
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"text": "Test-time self-adaptation To test the proposed test-time self-adaptation (TSA), we compared the CER performance of NANSY trained in three different configurations, 1. English (ENG), 2. English with TSA (ENG-TSA), 3. Multi-language (MUL). For every experiment, we iteratively updated the wav2vec feature 100 times. The results are shown in Fig. $\\triangledown$ Naturally, MUL generally showed better CER performance than other configurations. Interestingly, however, ENG-TSA sometimes showed similar or even better performance than MUL, which shows the effectiveness of the proposed TSA technique. ",
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"type": "image",
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"img_path": "images/6ab7510e2e4e43c70e414019e189f09a7485295bdba01aca11345f4b7fde516e.jpg",
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| 754 |
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"image_caption": [
|
| 755 |
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"Figure 6: The CER results on 10 languages (NL: Dutch, HU: Hungarian, FR: French, JP: Japanese, CH: Chinese, RU: Russian, FI: Finnish, GR: Greek, ES: Spanish, DE: German). "
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| 756 |
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| 758 |
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| 767 |
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"type": "text",
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| 768 |
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"text": "4.3 Voice conversion ",
|
| 769 |
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"text_level": 1,
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| 770 |
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"type": "text",
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"text": "NANSY can perform zero-shot voice conversion by simply passing the desired target speech utterance to the speaker embedding network. We first compared the English voice conversion performance of NANSY with recently proposed zero-shot voice conversion models. Next, we tested the multilingual voice conversion performance. Finally, we tested unseen language voice conversion for both seen speaker and unseen speaker targets. Note that in every voice conversion experiment, we shifted the median pitch of a source utterance to the median pitch of a target utterance by shifting the scope of Yingram. We measured naturalness with 5-scale mean opinion score (MOS [1-5]). Speaker similarity (SSIM $( \\% )$ ) were measured with a binary decision and uncertainty options, following $\\pmb { \\Vert 5 0 \\Vert }$ . For MOS and SSIM, we again used MTurk. The details of MOS and SSIM are shown in Appendix D. One of the crucial criteria for evaluating the quality of the converted samples is to check the intelligibility of them. Previous zero-shot voice conversion models, however, have only reported MOS or SSIM and have been negligent on assessing the intelligibility of the converted samples [36, 10, 51]. To this end, we report character error rate (CER $( \\% ) _ { , }$ ) between estimated characters of source and converted pairs. To estimate the characters from speech samples, we used google cloud ASR API. ",
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"text": "",
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"type": "text",
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"text": "Algorithm comparison Here, we report three source-to-target speaker conversion settings, 1. seen-to-seen (many-to-many, M2M), 2. unseen-to-seen (any-to-many, A2M), 3. unseen-to-unseen (any-to-any, A2A). For every setting, we considered 4 gender-to-gender combination, i.e., male-tomale $( { \\mathrm { m } } 2 { \\mathrm { m } } )$ , male-to-female (m2f), female-to-male $( \\mathrm { f } 2 \\mathrm { m } )$ , and female-to-female (f2f). For M2M setting, we randomly sampled 25 seen speakers from VCTK and randomly assigned 2 random speakers from VCTK, resulting in 200 $( = 2 5 { \\times } 2 { \\times } 4 )$ conversion pairs in total. For A2M setting, we randomly sampled 10 seen speakers from test-clean subset of LibriTTS and randomly assigned 2 random speakers from VCTK resulting in 80 $( = 1 0 \\times 2 \\times 4 )$ ) conversion pairs in total. For A2A setting, we randomly sampled 10 seen speakers from test-clean subset of LibriTTS and randomly assigned 2 random speakers from test-clean subset of LibriTTS resulting in 80 $( = 1 0 \\times 2 \\times 4 )$ conversion pairs in total. We trained three baseline models with official implementations - ${ \\mathrm { V Q V C } } +$ [51], AdaIN $[ \\mathbb { 1 0 } ]$ , AUTOVC $\\pmb { \\mathbb { B } } 6 \\|$ - using the same dataset and mel spectrogram configuration as NANSY. For a fair comparison, we used a pre-trained HiFi-GAN vocoder for every model. Table $\\triangledown$ shows that NANSY significantly outperforms previous models in terms of every evaluation measure. This implies that the proposed information perturbation approach does not suffer from the trade-off between CER and SSIM unlike information bottleneck approaches. The SSIM results for all possible gender-to-gender combinations are shown in Fig. 9 in Appendix C. ",
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"img_path": "images/220bd3e215a4369c45ee65c39b784536f46e4ac213179c01c8b9a6c03543a539.jpg",
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"table_caption": [],
|
| 815 |
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"table_footnote": [
|
| 816 |
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"Table 3: Evaluation results on English voice conversion. SRC and TGT denote, source and target, respectively. "
|
| 817 |
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],
|
| 818 |
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"table_body": "<table><tr><td>一</td><td></td><td>M2M</td><td></td><td></td><td>A2M</td><td></td><td></td><td>A2A</td><td></td></tr><tr><td></td><td></td><td>[CER[%] MOS[1-5]</td><td></td><td></td><td>]SSIM[%]|CER[%]MOS[1-5]</td><td></td><td></td><td>SSIM[%]|CER[%] MOS[1-5]</td><td>SSIM[%]</td></tr><tr><td>SRC as TGT</td><td>n/a</td><td>4.23 ± 0.05</td><td>0</td><td>n/a</td><td>4.28 ± 0.09</td><td>0.60</td><td>n/a</td><td>4.26 ± 0.07</td><td>0.25</td></tr><tr><td>TGT as TGT</td><td>n/a</td><td>4.32 ± 0.05</td><td>94.9</td><td>n/a</td><td>4.29 ± 0.05</td><td>92.4</td><td>n/a</td><td>4.27 ± 0.07</td><td>96.2</td></tr><tr><td>VQVC+</td><td>54.0</td><td>1.76 ± 0.05</td><td>54.5</td><td>74.7</td><td>1.73 ± 0.11</td><td>15.6</td><td>69.3</td><td>1.83 ± 0.09</td><td>13.8</td></tr><tr><td>AdaIN</td><td>62.9</td><td>2.22 ± 0.07</td><td>24.0</td><td>79.6</td><td>1.92 ± 0.12</td><td>18.1</td><td>59.3</td><td>2.12 ± 0.10</td><td>21.2</td></tr><tr><td>AUTOVC</td><td>31.7</td><td>3.41 ±0.06</td><td>47.3</td><td>36.1</td><td>2.74 ±0.11</td><td>33.2</td><td>28.2</td><td>2.59 ±0.08</td><td>23.3</td></tr><tr><td>NANSY</td><td>7.5</td><td>3.79 ± 0.07</td><td>91.4</td><td>7.6</td><td>3.73 ± 0.05</td><td>88.1</td><td>8.6</td><td>3.44 ± 0.07</td><td>64.6</td></tr></table>",
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"type": "text",
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"text": "Multilingual voice conversion We tested multilingual voice conversion performance with the model trained on the multilingual dataset, CSS10. We randomly sampled 50 samples for each language speaker from CSS10 and assigned random single target speaker from CSS10 for each source language, resulting in 500 $( = 5 0 \\times 1 0 )$ ) conversion pairs in total. The results in Table 4 show that the proposed framework can successfully perform multilingual voice conversion by training it with the multilingual dataset. However, there is still a room for improvement for multilingual voice conversion when comparing to the results in Table 3, where NANSY is just trained on English. ",
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"type": "text",
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| 840 |
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"text": "Unseen language voice conversion We tested the voice conversion performance on unseen source languages using NANSY trained on English. We tested the performance on two settings, 1. unseen source language (CSS10) to seen target voice (VCTK) and 2. unseen source language (CSS10) to unseen target voice (CSS10). For the first experiment, we randomly sampled 50 samples for each unseen language speaker from CSS10 and assigned random English target speakers from VCTK, resulting in 500 $( = 5 0 \\times 1 0 )$ conversion pairs in total. The second experiment was conducted identically to the multilingual voice conversion experiment setting. The results in Table $\\boxed { 5 }$ shows that NANSY can be successfully extended even to unseen language sources, although there was a decrease on SSIM compared to the results in Table 4 $( 6 9 . 5 \\% \\bar { } 6 \\bar { 1 } . 0 \\% )$ ). ",
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"type": "table",
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"img_path": "images/2f0250c67b62c70bf386286198c60813a90c48e9b34012ea923f88333fd6e4ef.jpg",
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"table_caption": [
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| 853 |
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"Table 4: The multilingual voice conversion results. The model was trained using only CSS10. "
|
| 854 |
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],
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| 855 |
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"table_footnote": [],
|
| 856 |
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"table_body": "<table><tr><td></td><td>|CER MOS</td><td>SSIM</td></tr><tr><td></td><td>TGT as TGT|n/a 4.23 ±0.06</td><td>98.0</td></tr><tr><td>NANSY</td><td>18.8 3.68 ±0.09</td><td>69.5</td></tr></table>",
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"img_path": "images/7a415393e7cc61843de35198d0130e77273eb1cda71ab16f649c8d55dd5f178e.jpg",
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"table_caption": [],
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| 870 |
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"Table 5: The voice conversion results on unseen language dataset, CSS10. The model was trained using only the English datasets. "
|
| 871 |
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],
|
| 872 |
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"table_body": "<table><tr><td></td><td colspan=\"3\">Seen Speaker</td></tr><tr><td></td><td>|CER MOS</td><td>SSIM|CER</td><td>MOS SSIM</td></tr><tr><td>TGT as TGT NANSY</td><td>n/a 4.24 ±0.07 14.8 3.75 ± 0.10</td><td>100 90.0</td><td>n/a 4.23 ± 0.06 92.0 15.6 3.76 ± 0.09 61.0</td></tr></table>",
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"text": "4.4 Pitch shift and time-scale modification ",
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"text_level": 1,
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"text": "To check the robustness of pitch shift (PS) and time-scale modification (TSM) performance of NANSY, we compared it with other robust algorithms, i.e., PSOLA 5 and WORLD vocoder [29, 28]. ",
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"text": "Pitch shift We tested PS performance with 5 semitone ranges, -6, -3, 0, 3, 6. The pitch was changed by shifting the scope of the proposed Yingram feature. Note that $\\mathbf { \\bar { \\rho } } _ { 0 } ,$ was used to check analysissynthesis performance. We randomly selected 20 samples for each semitone range from $10 \\%$ unseen utterances of VCTK. We evaluated the naturalness of speech samples with MOS on MTurk. The results in Table $6$ show that NANSY generally outperforms algorithms such as PSOLA and WORLD vocoder on PS task. ",
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"text": "Time-scale modification We tested TSM performance with 5 time-scale ratios, 1/2, 1/1.5, 1, 1.5, 2. The time-scale was modified by simply manipulating the hop length of the analysis features. Note that $\\cdot _ { 1 } \\cdot$ was used to check analysis-synthesis performance. We randomly selected 20 samples for each time-scale ratio from $10 \\%$ unseen utterances of VCTK. We evaluated the naturalness of speech samples with MOS on MTurk. The results in Table 7 show that NANSY achieved competitive performance on TSM compared to the well-established PSOLA and WORLD vocoder. ",
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"img_path": "images/c9059919ead5e483f7e80996f1fd99d72fc42b4019b948be7b795973cd8e2876.jpg",
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"table_caption": [],
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| 930 |
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"table_footnote": [
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| 931 |
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"Table 6: Pitch shift results. "
|
| 932 |
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| 933 |
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"table_body": "<table><tr><td></td><td>-6</td><td>-3</td><td>0</td><td>3</td><td>6</td></tr><tr><td>WORLD 28</td><td>3.433.53</td><td></td><td></td><td>33.853.63</td><td>3.53</td></tr><tr><td>PSOLA 四</td><td></td><td></td><td></td><td></td><td>3.633.553.933.753.48</td></tr><tr><td>NANSY</td><td>3.60 3.68</td><td></td><td>4.05</td><td>3.90</td><td>3.78</td></tr></table>",
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"type": "table",
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"img_path": "images/e2dc1319532906b37a461875e235d88d5075a05b1e9b83dae278794486e3c298.jpg",
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| 945 |
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"table_caption": [
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| 946 |
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"Table 7: Time-scale modification results. "
|
| 947 |
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],
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| 948 |
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"table_footnote": [],
|
| 949 |
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"table_body": "<table><tr><td></td><td>1/2</td><td>1/1.5 1</td><td>1.5</td><td>2</td></tr><tr><td>WORLD I28I</td><td>2.403.603.833.38</td><td></td><td></td><td>83.03</td></tr><tr><td>PSOLA 四</td><td>2.55 3.66 3.953.68</td><td></td><td></td><td>2.85</td></tr><tr><td>NANSY</td><td>2.453.63</td><td>33.983.70</td><td></td><td>2.93</td></tr></table>",
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"type": "text",
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"text": "5 Related Works ",
|
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"text": "Self-supervised representation learning and synthesis of speech There has been an increasing interest in the self-supervised learning methods within the machine learning and speech processing community. Oord et al. [31] first proposed to use noise contrastive estimation loss to train speech representations. Baevski et al. [4] extended this idea by integrating masked language modeling $\\mathbb { \\lVert 1 5 \\rVert }$ . Another popular self-supervised learning method for speech representation is to train a neural network by targeting multiple self-supervision tasks [34, 37]. Most recently, $\\pmb { \\Vert 3 5 \\Vert }$ used the discrete disentangled self-supervised representations to re-synthesize them into a waveform. Although using the discrete units has its own advantage in that it is disentangled with speaker information, we found that an inaccurate quantization process often leads to mispronounced samples, which is why we turned to use continuous representation as it can provide more accurate results on linguistic information. ",
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"text": "Zero-shot voice conversion Research on zero-shot voice conversion has been most actively conducted through the information bottleneck approach. Qian et al. $\\textcircled { \\lvert 3 6 \\rvert }$ proposed to perform zero-shot voice conversion by utilizing the pre-trained speaker recognition network and information bottleneck by carefully designing the bottleneck of an auto-encoder. Inspired by the success of style conversion in computer vision, Chou and Lee $\\mathbb { \\ m }$ also focused on restricting the information flow using instance normalization $\\pm \\ddagger { 4 }$ and adaptive instance normalization $\\mathbb { \\lVert 1 9 \\rVert }$ . Lastly, Wu et al. [51] used multiple vector quantization layers $\\bar { \\mathbb { B } } \\bar { 0 } \\bar { 1 }$ to restrict the information flow. ",
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"type": "text",
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"text": "Inductive bias for audio generation It has been shown that neural networks can be combined with traditional speech/sound production models for efficient and strong performance. One of the speech production models that has been integrated with neural networks is the source-filter model. By modeling source and filter components with deep networks, it has been used for applications such as vocoder [49, 23, 45] and acoustic feature generation $\\lVert 2 5 \\rVert$ . Furthermore, Engel et al. $\\mathbb { \\ m }$ proposed to integrate a harmonic plus noise model $\\bar { \\big \\| } \\bar { 3 8 } \\bar { \\big \\| }$ and neural networks to produce natural audio signal. ",
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"text": "Consistency learning Learning representations by augmenting the data has been one of the key ideas to leverage the performance of classification tasks [43, 52]. This shares the similar idea with the proposed information perturbation strategy in that the data is perturbed so that the neural network must learn to ignore the perturbations and learn the consistency from the data. However, the key difference between the consistency learning and the information perturbation is that the information perturbation method is designed for “generative” task and that it is the “decoder” (e.g., Generator) that is trained to selectively take the essential attributes to reconstruct the signal from the given perturbed representations. ",
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"text": "6 Conclusions and Discussion ",
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"text": "In this work, we proposed a neural analysis and synthesis framework (NANSY) that can perform zero-shot voice conversion, formant preserving pitch shift, and time-scale modification with a single model. The proposed model can be trained in a fully self-supervised manner, that is, it can be trained without any labeled data such as text or speaker information. We showed the effectiveness of the proposed information perturbation approach by showing the voice conversion results on various settings. Furthermore, we showed the effectiveness of the proposed TSA method by testing it on unseen languages, which shows the possibility of NANSY to be extended on low-resource languages. Although the proposed method empowers controllability over several attributes of a speech signal, it is still limited in terms of lacking controllability over linguistic information. As a future work, therefore, we would like to investigate on a hybrid approach that integrates text information as a side input so that the user can manipulate even the linguistic information in the speech signal. Finally, to prevent the proposed framework being used maliciously (e.g., voice phishing), it would be important to develop a detection algorithm that can discriminate a synthesized speech sample from a real speech sample. To examine the potential of such a detection system, we have tried using the trained Discriminator from the NANSY framework, which is expected to discriminate fake samples from real samples. We measured the accuracy of classifying 185 reconstructed samples and 185 ground truth samples. In addition, we measured the accuracy of classifying 560 voice conversion samples and 560 ground truth samples. The accuracy was $9 1 . 4 \\%$ on the reconstruction set and $9 5 . 5 \\%$ on the voice conversion set. This shows the possibility of Discriminator being used as a byproduct network to discriminate real speech samples from fake speech samples. However, we also found that Discriminator is prone to being deceived by the generated samples from other speech generative models as Discriminator was not jointly trained with those generative models. Therefore, we expect more robust synthesized speech detection algorithms to be developed in the future such as [48, 40, 9, 6]. ",
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"type": "text",
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"text": "Broader Impacts ",
|
| 1040 |
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"text_level": 1,
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| 1041 |
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"type": "text",
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| 1051 |
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"text": "The proposed NANSY framework shows that a generative model can benefit from self-supervised representations. By choosing proper domain specific “information perturbation” functions, we believe that one can achieve controllable generative modeling in a fully self-supervised way. The information perturbation training strategy may also be used for other modalities and facilitate self-supervised representation learning methods too. With the proposed training framework, one can manipulate various aspects of speech samples. Among the various controllabilities, it is rather obvious that the voice conversion technique can be misused and potentially harm other people. More concretely, there are possible scenarios where it is being used by random unidentified users and contributing to spreading fake news. In addition, it can raise concerns about biometric security systems based on speech. To mitigate such issues, the proposed system should not be released without a consent so that it cannot be easily used by random users with malicious intentions. That being said, there is still a potential for this technology to be used by unidentified users. As a more solid solution, therefore, we believe a detection system that can discriminate between fake and real speech should be developed. The preliminary results of the detection system is reported in section 6. ",
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"type": "text",
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| 1062 |
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"text": "Acknowledgments and Disclosure of Funding ",
|
| 1063 |
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"text_level": 1,
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| 1064 |
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"text": "This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) [NO.2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University)] ",
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"type": "text",
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"text": "References \n[1] Luc Ardaillon. Synthesis and expressive transformation of singing voice. PhD thesis, 11 2017. \n[2] Luc Ardaillon and Axel Roebel. Gci detection from raw speech using a fully-convolutional network. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 6739–6743. IEEE, 2020. \n[3] Bishnu S Atal and Suzanne L Hanauer. Speech analysis and synthesis by linear prediction of the speech wave. The journal of the acoustical society of America, 50(2B):637–655, 1971. \n[4] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael Auli. wav2vec 2.0: A framework for self-supervised learning of speech representations. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 12449–12460. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/2020/file/ 92d1e1eb1cd6f9fba3227870bb6d7f07-Paper.pdf. \n[5] Paul Boersma and Vincent Van Heuven. Speak and unspeak with praat. Glot International, 5 (9/10):341–347, 2001. \n[6] Clara Borrelli, Paolo Bestagini, Fabio Antonacci, Augusto Sarti, and Stefano Tubaro. Synthetic speech detection through short-term and long-term prediction traces. EURASIP Journal on Information Security, 2021(1):1–14, 2021. \n[7] Mingjian Chen, Xu Tan, Bohan Li, Yanqing Liu, Tao Qin, sheng zhao, and Tie-Yan Liu. Adaspeech: Adaptive text to speech for custom voice. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=Drynvt7gg4L. \n[8] Hyeong-Seok Choi, Changdae Park, and Kyogu Lee. From inference to generation: End-to-end fully self-supervised generation of human face from speech. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=H1guaREYPr. \n[9] Yeunju Choi, Youngmoon Jung, and Hoirin Kim. Neural mos prediction for synthesized speech using multi-task learning with spoofing detection and spoofing type classification. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 462–469. IEEE, 2021. \n[10] Ju-Chieh Chou and Hung-Yi Lee. One-Shot Voice Conversion by Separating Speaker and Content Representations with Instance Normalization. In Proc. Interspeech 2019, pages 664– 668, 2019. doi: 10.21437/Interspeech.2019-2663. URL http://dx.doi.org/10.21437/ Interspeech.2019-2663. \n[11] Alexis Conneau, Alexei Baevski, Ronan Collobert, Abdelrahman Mohamed, and Michael Auli. Unsupervised cross-lingual representation learning for speech recognition. arXiv preprint arXiv:2006.13979, 2020. \n[12] Yann N Dauphin, Angela Fan, Michael Auli, and David Grangier. Language modeling with gated convolutional networks. In International conference on machine learning, pages 933–941. PMLR, 2017. \n[13] Alain De Cheveigné and Hideki Kawahara. Yin, a fundamental frequency estimator for speech and music. The Journal of the Acoustical Society of America, 111(4):1917–1930, 2002. \n[14] Brecht Desplanques, Jenthe Thienpondt, and Kris Demuynck. ECAPA-TDNN: Emphasized Channel Attention, Propagation and Aggregation in TDNN Based Speaker Verification. In Proc. Interspeech 2020, pages 3830–3834, 2020. doi: 10.21437/Interspeech.2020-2650. URL http://dx.doi.org/10.21437/Interspeech.2020-2650. ",
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| 1477 |
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| 1479 |
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| 1480 |
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|
| 1481 |
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"type": "text",
|
| 1482 |
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"text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We described the goal and contribution of this paper and conducted experiments accordingly. \n(b) Did you describe the limitations of your work? [Yes] We explained the limitation of this work in the conclusion. \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] We discussed the potential negative societal impacts (e.g., voice phishing) of our work in the conclusion. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have read the ethics review guidelines. ",
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| 1492 |
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"type": "text",
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"text": "2. If you are including theoretical results... ",
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|
| 1503 |
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"type": "text",
|
| 1504 |
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"text": "(a) Did you state the full set of assumptions of all theoretical results? [No] We do not include theoretical results. \n(b) Did you include complete proofs of all theoretical results? [No] We do not include theoretical results. ",
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"type": "text",
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"text": "3. If you ran experiments... ",
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| 1525 |
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"type": "text",
|
| 1526 |
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"text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] The code is proprietary. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See section $4 . 1$ and Appendix A. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We have included the error bars for crowdsourcing evaluation. ",
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| 1536 |
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| 1537 |
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"text": "(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See section 4.1. ",
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| 1547 |
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"type": "text",
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"text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
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{
|
| 1558 |
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"type": "text",
|
| 1559 |
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"text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We cited all three datasets we used for experiments. \n(b) Did you mention the license of the assets? [Yes] See section 4.1. \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] We did not curate/release any new assets. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We cited the paper of the datasets we are using in which they explain all the details regarding speaker recruitment. \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] We used speaker labels to split the datasets as described in section 4.1. ",
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| 1569 |
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| 1570 |
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"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
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|
| 1580 |
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"type": "text",
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| 1581 |
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"text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [Yes] We have attached the screenshots of MTurk instructions in Appendix D. \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [Yes] This work is approved by IRB (IRB No. 2105/004-008). We have announced that the de-identified information such as worker ID will be collected through MTurk instructions. \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [Yes] It is shown in Appendix D. ",
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]
|
parse/train/Aw96fN64soV/Aw96fN64soV_middle.json
ADDED
|
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|
|
parse/train/Aw96fN64soV/Aw96fN64soV_model.json
ADDED
|
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|
|
|
parse/train/B1hYRMbCW/B1hYRMbCW.md
ADDED
|
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| 1 |
+
# ON THE REGULARIZATION OF WASSERSTEIN GANS
|
| 2 |
+
|
| 3 |
+
Henning Petzka∗ Fraunhofer Institute IAIS, Sankt Augustin, Germany henning.petzka@gmail.com
|
| 4 |
+
|
| 5 |
+
Asja Fischer∗& Denis Lukovnikov Department of Computer Science, University of Bonn, Germany asja.fischer@gmail.com lukovnik@cs.uni-bonn.de
|
| 6 |
+
|
| 7 |
+
# ABSTRACT
|
| 8 |
+
|
| 9 |
+
Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a different metric, but thereby introduce a Lipschitz constraint into the optimization problem. A simple way to enforce the Lipschitz constraint on the class of functions, which can be modeled by the neural network, is weight clipping. Augmenting the loss by a regularization term that penalizes the deviation of the gradient norm of the critic (as a function of the network’s input) from one, was proposed as an alternative that improves training. We present theoretical arguments why using a weaker regularization term enforcing the Lipschitz constraint is preferable. These arguments are supported by experimental results on several data sets.
|
| 10 |
+
|
| 11 |
+
# 1 INTRODUCTION
|
| 12 |
+
|
| 13 |
+
General adversarial networks (GANs) (Goodfellow et al., 2014) are a class of generative models that have recently gained a lot of attention. They are based on the idea of defining a game between two competing neural networks (NNs): a generator and a classifier (or discriminator). While the classifier aims at distinguishing generated from real data, the generator tries to generate samples which the classifier can not distinguish from the ones from the empirical distribution. Realizing the potential behind this new approach to generative models, more recent contributions focused on the stabilization of training, including ensemble methods (Tolstikhin et al., 2017), improved network structure (Radford et al., 2015; Salimans et al., 2016) and theoretical improvements (Nowozin et al., 2016; Salimans et al., 2016; Arjovsky & Bottou, 2017; Chen et al., 2016) that helped to successfully model complex distributions using GANs.
|
| 14 |
+
|
| 15 |
+
It was proposed by Arjovsky et al. (2017) to train generator and discriminator networks by minimizing the Wasserstein-1 distance, a distance with properties superior to the Jensen-Shannon distance (used in the original GAN) in terms of convergence. Accordingly, this version of GAN was called Wasserstein GAN (WGAN). The change of metric introduces a new minimization problem, which requires the discriminator function to lie in the space of 1-Lipschitz functions. In the same paper, the Lipschitz constraint was guaranteed by performing weight clipping, i.e., by constraining the parameters of the discriminator NN to be smaller than a given value in magnitude. An improved training strategy was proposed by Gulrajani et al. (2017) based on results from optimal transport theory (see Villani, 2008). Here, instead of clipping weights, the loss gets augmented by a regularization term that penalizes any deviation of the norm of the gradient of the critic function (with respect to its input) from one.
|
| 16 |
+
|
| 17 |
+
We review these results and present both theoretical considerations and empirical results, leading to the proposal of a less restrictive regularization term for WGANs.1 More precisely, our contributions are as follows:
|
| 18 |
+
|
| 19 |
+
• We review the arguments that the regularization technique proposed by Gulrajani et al. (2017) is based on and make the following two observations: (i) The regularization strategy requires training samples and generated samples to be drawn from a certain joint distribution. In practice, however, samples are drawn independently from their marginals. (ii) The arguments further assume the discriminator to be differentiable. We explain why both can be harmful for training.
|
| 20 |
+
|
| 21 |
+
• We propose a less restrictive regularization term and present empirical results strongly supporting our theoretical considerations.
|
| 22 |
+
|
| 23 |
+
# 2 OPTIMAL TRANSPORT
|
| 24 |
+
|
| 25 |
+
We will require the notion of a coupling of two probability distributions. Although a coupling can be defined more generally, we state the definition in the setting of our interest, i.e., we consider all spaces involved to equal $\mathbb { R } ^ { n }$ .
|
| 26 |
+
|
| 27 |
+
Definition 1. Let $\mu$ and $\nu$ be two probability distributions on $\mathbb { R } ^ { n }$ . A coupling $\pi$ of $\mu$ and $\nu$ is a probability distribution on $\mathbb { R } ^ { n } \times \mathbb { R } ^ { n }$ such that $\pi ( A , \mathbb { R } ^ { n } ) = \mu ( A )$ and $\pi ( \mathbb { R } ^ { n } , A ) = \nu ( A )$ for all measurable sets $A \subseteq \mathbb { R } ^ { n }$ . The set of all couplings of $\mu$ and $\nu$ is denoted by $\Pi ( \mu , \nu )$ .
|
| 28 |
+
|
| 29 |
+
The following theorem plays a central role in the theory of optimal transport (OT) and is known as the Kantorovich duality. Note, that the presented theorem is a less general, but to our needs adapted version of Theorem 5.10 from Villani (2008).2 A proof of how to derive our version from the referenced one can be found in Appendix C.1. We will denote by $\mathcal { L } i p _ { 1 }$ the set of all 1-Lipschitz functions, i.e., the set of all functions $f$ such that $f ( y ) - f ( x ) \leq | | { \dot { x } } - { \bar { y } } | | _ { 2 }$ for all $x , y$ .
|
| 30 |
+
|
| 31 |
+
Theorem 1 (Kantorovich). Let $\mu$ and $\nu$ be two probability distributions on $\mathbb { R } ^ { n }$ such that $\begin{array} { r } { \int _ { \mathbb { R } ^ { n } } | | x | | _ { 2 } d \mu ( x ) < \infty } \end{array}$ and $\begin{array} { r } { \int _ { \mathbb { R } ^ { n } } | | x | | _ { 2 } d \nu ( x ) < \infty } \end{array}$ . Then
|
| 32 |
+
|
| 33 |
+
$$
|
| 34 |
+
\operatorname* { m i n } _ { \pi \in \Pi ( \mu , \nu ) } \int _ { \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } } | | x - y | | _ { 2 } d \pi ( x , y ) = \operatorname* { m a x } _ { f \in \mathcal { L } i p _ { 1 } } \left( \int _ { \mathbb { R } ^ { n } } f ( x ) d \mu ( x ) - \int _ { \mathbb { R } ^ { n } } f ( x ) d \nu ( x ) \right) \ .
|
| 35 |
+
$$
|
| 36 |
+
|
| 37 |
+
In particular, both minimum and maximum exist.
|
| 38 |
+
|
| 39 |
+
(ii) The following two statements are equivalent:
|
| 40 |
+
|
| 41 |
+
(a) $\pi ^ { * }$ is an optimal coupling (minimizing the value on the left hand side of (1)). (b) Any optimal function $f ^ { \ast } \in \mathcal { L } i p _ { 1 }$ (at which the maximum is attained for the right hand side of (1)) satisfies that for all $( x , y )$ in the support of $\pi ^ { * }$ : $f ^ { * } ( x ) - f ^ { * } ( y ) = | | x - y | | _ { 2 }$ .
|
| 42 |
+
|
| 43 |
+
The field of OT offers several approaches to the computation of optimal couplings. To speed up the computation of an optimal coupling, Cuturi (2013) introduced a regularized version of the primal problem in which an entropic term $E ( \pi )$ is added leading to the minimization of $\begin{array} { r } { \int _ { \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } } | | x \stackrel { \cdot } { - } y | | _ { 2 } \ d \pi ( x , y ) + \epsilon E ( \pi ) } \end{array}$ , with regularization parameter $\epsilon$ . Regularized OT was generalized by Dessein et al. (2016) to a more general class of regularization terms $\Omega ( \pi )$ . As we will discuss in Section 5, the learning algorithm we propose in this paper has connections to the approach using $\begin{array} { r } { \Omega ( \pi ) = \int \left( \frac { \mathrm { d } \pi ( x , y ) } { \mathrm { d } \mu ( x ) \mathrm { d } \nu ( y ) } \right) ^ { 2 } \mathrm { d } \mu ( x ) \mathrm { d } \nu ( y ) } \end{array}$ . By Blondel et al. (2017), this leads to the dual problem given by
|
| 44 |
+
|
| 45 |
+
$$
|
| 46 |
+
\operatorname* { s u p } _ { f , g } \left\{ \mathbb { E } _ { x \sim \mu } [ f ( x ) ] - \mathbb { E } _ { y \sim \nu } [ g ( y ) ] - \frac { 4 } { \epsilon } \int \int \operatorname* { m a x } \left\{ 0 , \left( f ( x ) - g ( y ) - | | x - y | | _ { 2 } \right) \right\} ^ { 2 } \mathrm { d } \mu ( x ) \mathrm { d } \nu ( y ) \right\} \ .
|
| 47 |
+
$$
|
| 48 |
+
|
| 49 |
+
# 3 WASSERSTEIN GANS
|
| 50 |
+
|
| 51 |
+
Formally, given an empirical distribution $\mu$ , a class of generative distributions $\nu$ over some space $\mathcal { X }$ , and a class of discriminators $d : \mathcal { X } [ 0 , 1 ]$ , GAN training (Goodfellow et al., 2014) aims at solving the optimization problem given by $\begin{array} { r } { \operatorname* { m i n } _ { \nu } \operatorname* { m a x } _ { d } \mathbb { E } _ { x \sim \mu } [ \log ( d ( x ) ) ] + \mathbb { E } _ { y \sim \nu } [ \log ( 1 - d ( y ) ) ] } \end{array}$ . 3 In practice, the parameters of the generator and the discriminator networks are updated in an alternating fashion based on (several steps) of stochastic gradient descent. The discriminator thereby tries to assign a value close to zero to generated data points and values close to one to real data points. As an opposing agent, the generator aims to produce data where the discriminator expects to see real data. Theorem 1 by Goodfellow et al. (2014) shows that, if the optimal discriminator is found in each iteration, minimization of the resulting loss function of the generator leads to minimization of the Jensen-Shannon (JS) divergence. Instead of minimizing the JS divergence, Arjovsky et al. (2017) proposed to minimize the Wasserstein-1 distance, also known as Earth-Mover (EM) distance, which is defined for any Polish space $( M , c )$ and probability distributions $\mu$ and $\nu$ on $M$ by
|
| 52 |
+
|
| 53 |
+
$$
|
| 54 |
+
W ( \mu , \nu ) = \operatorname* { i n f } _ { \pi \in \Pi ( \mu , \nu ) } \int _ { M \times M } c ( x , y ) d \pi ( x , y ) .
|
| 55 |
+
$$
|
| 56 |
+
|
| 57 |
+
From the Kantorovich duality (see Theorem 1, (i)) it follows that, in the special case we are considering, the infimum is attained and, instead of computing this minimum in Equation (3), the Wasserstein-1 distance can also be computed as
|
| 58 |
+
|
| 59 |
+
$$
|
| 60 |
+
W ( \mu , \nu ) = \operatorname* { m a x } _ { f \in \mathcal { L } i p _ { 1 } } \mathbb { E } _ { x \sim \mu } [ f ( x ) ] - \mathbb { E } _ { y \sim \nu } [ f ( y ) ] \mathrm { ~ , ~ }
|
| 61 |
+
$$
|
| 62 |
+
|
| 63 |
+
where the maximum is taken over the set of all 1-Lipschitz functions $\mathcal { L } i p _ { 1 }$
|
| 64 |
+
|
| 65 |
+
Thus, the WGAN objective is to solve
|
| 66 |
+
|
| 67 |
+
$$
|
| 68 |
+
\operatorname* { m i n } _ { \nu } \operatorname* { m a x } _ { f \in \mathcal { L } i p _ { 1 } } \mathbb { E } _ { x \sim \mu } [ f ( x ) ] - \mathbb { E } _ { y \sim \nu } [ f ( y ) ] \mathrm { ~ , ~ }
|
| 69 |
+
$$
|
| 70 |
+
|
| 71 |
+
which can be achieved by alternating gradient descent updates for the generating network $\nu$ and the 1-Lipschitz function $f$ (also modeled by a NN), just as in the case of the original GAN. The objective of the generator is still to generate real-looking data points and is led by function values of $f$ that plays the role of an appraiser (or critic). The appraiser’s goal is to assign a value of confidence to each data point, which is as low as possible on generated data points and as high as possible on real data. The confidence value it can assign is bounded by a constraint of similarity, where similarity is measured by the distance of data points. This can be motivated by the idea that similar points should have similar values of confidence for being real. The new role of the critic helps to solve convergence problems, but the interpretation of its value as classifying real (close to 1) and fake data (close to 0) is lost. We refer to Appendix A for a detailed discussion.
|
| 72 |
+
|
| 73 |
+
# 4 IMPROVED TRAINING OF WGANS
|
| 74 |
+
|
| 75 |
+
Modeling the WGAN critic function by a NN raises the question on how to enforce the 1-Lipschitz constraint of the objective in Equation (5). As proposed by Arjovsky et al. (2017) a simple way to restrict the class of functions $f$ that can be modeled by the NN to $\alpha$ -Lipschitz continuous functions (for some $\alpha$ ) is to perform weight clipping, i.e. to enforce the parameters of the network not to exceed a certain value $c _ { \operatorname* { m a x } } > 0$ in absolute value. As the authors note, this is not a good but simple choice. We further demonstrate this in Appendix B by proving (for a standard NN architecture) that, using weight clipping, the optimal function is in general not contained in the class of functions modeled by the network.
|
| 76 |
+
|
| 77 |
+
Recently, an alternative to weight clipping was proposed by Gulrajani et al. (2017). The basic idea is to augment the WGAN loss by a regularization term that penalizes the deviation of the gradient norm of the critic with respect to its input from one (leading to a variant referred to as WGAN-GP, where GP stands for gradient penalty). More precisely, the loss of the critic to be minimized is then given by
|
| 78 |
+
|
| 79 |
+
$$
|
| 80 |
+
\begin{array} { r } { \mathbb { E } _ { y \sim \nu } [ f ( y ) ] - \mathbb { E } _ { x \sim \mu } [ f ( x ) ] + \lambda \mathbb { E } _ { \hat { x } \sim \tau } [ ( | | \nabla f ( \hat { x } ) | | _ { 2 } - 1 ) ^ { 2 } ] \ , } \end{array}
|
| 81 |
+
$$
|
| 82 |
+
|
| 83 |
+
where $\tau$ is the distribution of $\hat { x } = t x + ( 1 - t ) y$ for $t \sim U [ 0 , 1 ]$ and $x \sim \mu , y \sim \nu$ being a real and a generated sample, respectively. The regularization term is derived based on the following result.
|
| 84 |
+
|
| 85 |
+
Proposition 1. Let $\mu$ and $\nu$ be two probability distributions on $\mathbb { R } ^ { n }$ . Let $f ^ { * }$ be an optical critic, leading to the maximum $\begin{array} { r } { \operatorname* { m a x } _ { f \in \mathcal { L } i p _ { 1 } } \bar { \int } _ { \mathbb { R } ^ { n } } f ( x ) \mathop { d \mu ( x ) } - \int _ { \mathbb { R } ^ { n } } f ( x ) \mathop { d \nu } ( x ) } \end{array}$ , and let $\pi ^ { * }$ be an optimal coupling with respect to $\begin{array} { r } { \operatorname* { m i n } _ { \pi \in \Pi ( \mu , \nu ) } \int _ { \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } } | | x - y | | _ { 2 } d \pi ( x , y ) } \end{array}$ . If $f ^ { * }$ is differentiable and $x _ { t } =$ $t x + ( 1 - t ) y$ for $0 \leq t \leq 1$ , it holds that $\begin{array} { r } { \mathbb { P } _ { ( x , y ) \sim \pi ^ { * } } \Big [ ( \nabla f ^ { * } ( x _ { t } ) = \frac { y - x _ { t } } { | | y - x _ { t } | | } ) \Big ] = 1 } \end{array}$ . This in particular implies, that the norms of the gradients are one $\pi ^ { * }$ -almost surely on such points $x _ { t }$ .
|
| 86 |
+
|
| 87 |
+
For the convenience of the reader, we provide a simple argument for obtaining this result in Appendix C.2.
|
| 88 |
+
|
| 89 |
+
Note, that Proposition 1 holds only when $f ^ { * }$ is differentiable and $x$ and $y$ are sampled from the optimal coupling $\pi ^ { * }$ . However, sampling independently from the marginal distributions $\mu$ and $\nu$ very likely results in points $( x , y )$ that lie outside the support of $\pi ^ { * }$ . Furthermore, the optimal cost function $f ^ { * }$ does not need not to be differentiable everywhere. These two points will be discussed in more detail in the following subsections.
|
| 90 |
+
|
| 91 |
+
# 4.1 SAMPLING FROM THE MARGINALS INSTEAD OF THE OPTIMAL COUPLING
|
| 92 |
+
|
| 93 |
+
Observation 1. Suppose $f ^ { * } \in \mathcal L i p _ { 1 }$ is an optimal critic function and $\pi ^ { * }$ the optimal coupling determined by the Kantorovich duality in Theorem 1. Then $| f ^ { * } ( y ) - f ^ { * } ( x _ { t } ) | = | | x _ { t } - y | | _ { 2 }$ on the line $x _ { t } = t x + ( 1 - t ) y$ , $0 \leq t \leq 1$ , for $( x , y )$ sampled from $\pi ^ { * }$ , but not necessarily on the lines connecting an arbitrary pair of a real and $a$ generated data point, i.e. arbitrary $x \sim \mu$ and $y \sim \nu$ .
|
| 94 |
+
|
| 95 |
+
Consider the examples in Figure 1, where every $\mathrm { X }$ denotes a sample from the generator and every O a real data sample. Optimal couplings $\pi ^ { * }$ are indicated in red, and values of an optimal critic function are indicated in blue (optimality is shown in Appendix A.1).
|
| 96 |
+
|
| 97 |
+
|
| 98 |
+
Figure 1: A one (left) and a two (right) dimensional example showing that $f ^ { * } ( \mathbf { O } )$ - $f ^ { * } ( \mathbf { X } ) { = } | \mathbf { O } { - } \mathbf { X } |$ only holds for coupled pairs $( \mathbf { X } , 0 ) \sim \pi ^ { * }$ .
|
| 99 |
+
|
| 100 |
+
In the one-dimensional example on the left, the left-most X and the right-most O satisfy $f ^ { * } ( 0 ) -$ $f ^ { * } ( \mathbf { \boldsymbol { X } } ) = \frac { 1 } { 7 } | 0 - \mathbf { \boldsymbol { X } } | \neq | 0 - \dot { \mathbf { \boldsymbol { X } } | }$ , illustrating that the basis for the derivation of the condition, that the norm of the gradient equals one between generated and real points, only holds for points sampled from the optimal coupling. Note, while here the gradient is still of norm 1 almost everywhere, this does not necessarily hold in higher dimensions, where not all points lie on a line between some pair of points sampled from $\pi ^ { * }$ . This is exemplified for two dimensions on the right side of Figure 1, where blue numbers with $a \in \mathbb { R }$ denote the values of an optimal critic function at these points (the values at these points is all that matters). Without loss of generality we can assume the value at position $( 1 , 2 )$ to be zero, taking into account that an optimal critic function remains optimal under addition of an arbitrary constant. Since the Lipschitz constraint of $f ^ { * }$ must be satisfied, we get $1 - a \leq { \sqrt { 2 } }$ and $a + 1 \le { \sqrt { 2 } }$ . Therefore $a \in [ \bar { 1 } - \sqrt { 2 } , \sqrt { 2 } - 1 ]$ and one of the inequalities of the Lipschitz constraint must be strict.
|
| 101 |
+
|
| 102 |
+
# 4.2 DIFFERENTIABILITY OF THE CRITIC
|
| 103 |
+
|
| 104 |
+
Observation 2. The assumption of differentiability of the optimal critic is not valid at points of interest.
|
| 105 |
+
|
| 106 |
+
Consider the example of two discrete probability distributions and its optimal critic function $f ^ { * }$ shown on the left in Figure 2. We can see that the indicated function $f ^ { * } ( x ) = 1 - | x | \in \mathcal { L } i p _ { 1 }$ is optimal as it leads to an equality in the equation of the Kantorovich dual. (Also, it is the only continuous function, up to a constant, that realizes $f ^ { * } ( x ) - f ^ { * } ( y ) = | y - x |$ for coupled points $( x , y )$ .) However, it is not differentiable at 0.
|
| 107 |
+
|
| 108 |
+
|
| 109 |
+
Figure 2: Non-differentiable optimal critic functions $\mathrm { f ^ { * } }$ (shown in blue). Left: For two discrete distributions: Circles and crosses belong to samples from the empirical distribution and the generative model, respectively. An approximating differentiable function is shown in green. Right: For two continuous distributions: The empirical distribution $\mu$ is shown in gray, the generative distribution $\nu$ is shown in green.
|
| 110 |
+
|
| 111 |
+
The counterexample can be made continuous by considering the points as the center points of Gaussians, as illustrated on the right in Figure 2. This is formalized by the following proposition showing that the critic indicated in blue is indeed optimal for the depicted gray Gaussian of real data and the green mixture of two Gaussians of generated data.
|
| 112 |
+
|
| 113 |
+
Proposition 2. Let $\mu = \mathcal { N } ( 0 , 1 )$ be a normal distribution centered around zero and $\nu = \nu _ { - 1 } + \nu _ { 1 }$ be a mixture of the two normal distributions $\begin{array} { r } { \nu _ { - 1 } = \frac { 1 } { 2 } \mathcal { N } ( - 1 , 1 ) } \end{array}$ and $\begin{array} { r } { \nu _ { 1 } = \frac { 1 } { 2 } \mathcal { N } ( 1 , 1 ) } \end{array}$ over the real line. If $\mu$ describes the distribution of real data and $\nu$ describes the distribution of the generative model, then an optimal critic function is given by $\phi ^ { * } ( x ) = - | x |$ .
|
| 114 |
+
|
| 115 |
+
The proof can be found in Appendix C.3.
|
| 116 |
+
|
| 117 |
+
The issue with non-differentiability can be generalized to higher-dimensional spaces based on the observation that an optimal coupling is in general not deterministic. Deterministic couplings are particularly nice in the sense that they allow a transport plan assigning each point $x$ from one distribution deterministically to a point $y$ of the other distribution, without having to split any masses (the search for deterministic optimal couplings is called the Monge problem). However, in a lot of settings no deterministic coupling exists. The notion of a deterministic coupling is formalized in the following definition.
|
| 118 |
+
|
| 119 |
+
Definition 2. Let $( X , \mu )$ and $( Y , \nu )$ be two probability spaces. A coupling $\pi \in \Pi ( \mu , \nu )$ is called deterministic if there is a measurable function $\rho : X Y$ such that su $\eta ( \pi ) \subseteq \{ ( x , \rho ( x ) ) | x \in X \}$ .
|
| 120 |
+
|
| 121 |
+
We can now formulate the following observation.
|
| 122 |
+
|
| 123 |
+
Observation 3. Suppose $\pi ^ { * }$ is a non-deterministic optimal coupling between two probability distributions over $\mathbb { R } ^ { n }$ so that there exist points $( x , y )$ and $( x , y ^ { \prime } )$ in $s u p p ( \pi ^ { * } )$ . Suppose further that there is no $\lambda > 0$ with $( y - x ) = \lambda \cdot ( y ^ { \prime } - x )$ (in particular this implies $y \ne y ^ { \prime }$ ). Then any optimal critic function $f ^ { * }$ is not differentiable at $x$ .
|
| 124 |
+
|
| 125 |
+
The arguments can be found in Appendix C.5.
|
| 126 |
+
|
| 127 |
+
In practice, where the optimal critic is approximated by a NN, the situation is slightly different: A function modeled by an NN is (almost) everywhere differentiable (depending on the activation functions). By the Stone-Weierstrass theorem, on compact sets, we can approximate any (Lipschitz-)continuous function by differentiable functions uniformly. Nevertheless, it seems to be a strong constraint on an approximating function to have a gradient of norm one in the neighborhood of a non-differentiability (cf. Figure 2 (a)). Therefore, we argue – in contrast to the argumentation of Gulrajani et al. (2017) – that the gradient should not be assumed to equal one for arbitrary points on the line between an arbitrary real point $x$ and a generated point $y$ .
|
| 128 |
+
|
| 129 |
+
# 5 HOW TO REGULARIZE WGANS
|
| 130 |
+
|
| 131 |
+
In the following, we will discuss how the regularization of WGANs can be improved.
|
| 132 |
+
|
| 133 |
+
Penalizing the violation of the Lipschitz constraint. For the critic function, we have nothing more at hand than the inequality of the Lipschitz-constraint. Moreover (as shown in Lemma 1 in the Appendix) the exhaustion of the Lipschitz constant is automatic by maximizing the objective function. Therefore, a natural choice of regularization is to penalize the given constraint directly, i.e., sample two points $x \sim \mu$ and $y \sim \nu$ from the empirical and the generated distribution respectively and add the regularization term
|
| 134 |
+
|
| 135 |
+
$$
|
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+
\left( \operatorname* { m a x } \left\{ 0 , { \frac { | f ( x ) - f ( y ) | } { | | x - y | | _ { 2 } } } - 1 \right\} \right) ^ { 2 }
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$$
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to the cost function. (We square to penalize larger deviations more than smaller ones.) Note the similarity of the regularization term to the squared Hinge loss, which is also used to turn a hard constraint into a soft one in the optimization problem connected to support vector machines.
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Alternatively, since the NN generates (almost everywhere) differentiable functions, we can penalize whenever gradient norms are strictly larger than one, an option referred to as “one-sided penalty” and shortly discussed as an alternative to penalizing any deviation from one by Gulrajani et al. $( 2 0 1 7 ) ^ { 4 }$ . Note that enforcing the gradient to be smaller than one in norm has the advantage that we penalize when the partial derivative has norm $> 1$ into the direction of steepest descent. Hence, all partial derivatives are implicitly enforced to be bounded in norm by one, too. At the same time, enforcing $\leq 1$ for the gradient of smooth approximating functions is not an unreasonable constraint even at points of non-differentiability. For these reasons we suggest to add the regularization term $\big ( \operatorname* { m a x } { \{ 0 , \lvert \lvert \nabla f ( \hat { x } ) \rvert \rvert - 1 \} } \big ) ^ { 2 }$ to the cost function. Different ways of sampling the point $\hat { x }$ are analyzed in Appendix D.4. Thus, our proposed method (WGAN-LP, where LP stands for Lipschitz penalty) alternates between updating the discriminator to minimize
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$$
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\begin{array} { r } { \mathbb { E } _ { y \sim \nu } [ f ( y ) ] - \mathbb { E } _ { x \sim \mu } [ f ( x ) ] + \lambda \mathbb { E } _ { \hat { x } \sim \tau } [ ( \operatorname* { m a x } \left\{ 0 , | | \nabla f ( \hat { x } ) | | - 1 \right\} ) ^ { 2 } ] \ , } \end{array}
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$$
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(where $\tau$ depends on the concrete sampling strategy chosen) and updating the generator network modeling $\nu$ to minimize $- \mathbb { E } _ { y \sim \nu } [ f ( y ) ]$ using gradient descent.
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The connection to regularized optimal transport. Consider Equation (2) of regularized OT. For a hard constraint $f ( x ) - g ( y ) \leq | | x - y | | _ { 2 }$ , one can attain the supremum over $\bar { \mathbb { E } } _ { x \sim \mu } [ f ( x ) ] -$ $\mathbb { E } _ { y \sim \nu } [ g ( y ) ] - 0$ by setting $f ( x ) = \operatorname* { i n f } _ { y } g ( y ) + | | x - y | | _ { 2 } = g ( x )$ and subsequently maximize over one function only. Taking the advantage of dealing with a single function as a motivation, one may similarly replace $f = g$ in Equation 2, which uses a soft constraint (even though this can now only approximate the supremum). This leads to an objective of minimizing
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$$
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\mathbb { E } _ { y \sim \nu } [ f ( y ) ] - \mathbb { E } _ { x \sim \mu } [ f ( x ) ] + \frac { 4 } { \epsilon } \int \int \operatorname* { m a x } \big \{ 0 , ( f ( x ) - f ( y ) - | | x - y | | _ { 2 } ) \big \} ^ { 2 } \mathrm { d } \mu ( x ) \mathrm { d } \nu ( y )
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$$
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that, similarly to Equation (7), softly penalizes whenever $f ( x ) - f ( y ) > | | x - y | | _ { 2 }$ for a real sample $x$ and a generated sample $y$ . It is noteworthy that to justify the replacement $f \ = \ g$ one would require a high regularization parameter $\lambda = \frac { 4 } { \epsilon }$ of the dual problem, which corresponds to a low regularization of the primal problem.
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Dependence on the regularization hyperparameter $\lambda$ . Let $\mathcal { L } _ { \lambda } ^ { G P }$ and $\mathcal { L } _ { \lambda } ^ { L P }$ denote the infimums of the regularized losses over a class of (differentiable) critic functions $f$ from Equation (6) (WGANGP) and Equation (8) (WGAN-LP) respectively. For the comparison of these optimal losses we have the following result (proof in Appendix C.4).
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# Proposition 3.
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$$
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\begin{array} { r } { \mathcal { L } _ { \lambda } ^ { L P } \le \mathcal { L } _ { \lambda } ^ { G P } \le \mathcal { L } _ { \lambda } ^ { L P } + \lambda } \end{array}
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$$
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In particular, for small $\lambda$ the optimal scores approximately agree. On the other hand, increasing $\lambda$ strengthens the soft constraints, which means that the theoretical observations from Section 4 become more pertinent with growing $\lambda$ . Our experiments show exactly the behavior that WGANLP and WGAN-GP perform very similarly for small $\lambda$ , while WGAN-LP performs much better for larger values of $\lambda$ and its performance is much less dependent on the choice of hyperparameter $\lambda$ .
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A more general view. The Kantorovich duality theorem holds in a quite general setting. For example, a different metric can be substituted for the Euclidean distance $| | \cdot | | _ { 2 }$ . Taking $| | \cdot | | _ { 2 } ^ { \bar { p } }$ for a different natural number $p$ for example leads to the minimization of the Wasserstein distance of order $p$ (i.e., the Wasserstein- $p$ distance). Based on the dual problem to the computation of the Wasserstein distance of order $p$ (as given by the Kantorovich duality theorem) we still need to maximize Equation (5) with the only difference that 1-Lipschitz-continuity is now measured with respect to $| | \cdot | | _ { 2 } ^ { p }$ . For our training method this entails that the only modification to make is to use the regularization term given by (7), where the Euclidean distance is replaced by the metric of interest. We provide experimental results for the Wasserstein-2 distance in Appendix D.5.
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Recently, by Bellemare et al. (2017), the Wasserstein distance was replaced by the energy distance 5. For the training of Cramer GANs, the authors apply the GP-penalty term proposed by Gulrajani et al. (2017). We expect that using the LP-penalty term instead is also beneficial for Cramer GANs.
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# 6 EXPERIMENTS
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We perform several experiments on three toy data sets, 8Gaussians, 25Gaussians, and Swiss Roll 6, to compare the effect of different regularization terms. More specifically, we compare the performance of WGAN-GP and WGAN-LP as described in Equations (6) and (8) respectively, where the penalty was applied to points randomly sampled on the line between the training sample $x$ and the generated sample $y$ . Other sampling methods are discussed in Appendix D.4.
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Both, the generator network and the critic network, are simple feed-forward NNs with three hidden Leaky ReLU layers, each containing 512 neurons, and one linear output layer. The dimensionality of the latent variables of the generator network was set to two. During training, 10 critic updates are performed for every generator update, except for the first 25 generator updates, where the critic is updated 100 times for each generator update in order to get closer to the optimal critic in the beginning of training. Both networks were trained using RMSprop (Tijmen & Hinton, 2012) with learning rate $5 \cdot 1 0 ^ { - 5 }$ and a batch size of 256.
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To see whether our findings on toy data sets can be transferred to real world settings, we trained bigger WGAN-GPs and WGAN-LPs on CIFAR-10 as it is described below. Code for the reproduction of our results is available under https://github.com/lukovnikov/improved_wgan_ training .
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Level sets of the critic. A qualitative way to evaluate the learned critic function for a twodimensional data set is by displaying its level sets, as it was done by Gulrajani et al. (2017) and Kodali et al. (2017). The level sets after 10, 50, 100 and 1000 training iterations of a WGAN trained with the GP and LP penalty on the Swiss Roll data set are shown in Figure 3. Similar experimental results for the 8Gaussians and 25Gaussian data sets can be found in Appendix D.1.
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It becomes clear that with a penalty weight of $\lambda = 1 0$ , which corresponds to the hyperparameter value suggested by Gulrajani et al. (2017), the WGAN-GP does neither learn a good critic function nor a good model of the data generating distribution. With a smaller regularization parameter, $\lambda = 1$ , learning is stabilized. However, with the LP-penalty a good critic is learned even with a high penalty weight in only a few iterations and the level sets show higher regularity. Training a WGAN-LP with lower penalty weight led to equivalent observations (results not shown). We also experimented with much higher values for $\lambda$ , which led to almost the same results as for $\lambda = 1 0$ , which emphasizes that LP-penalty based training is less sensitive to the choice of $\lambda$ .
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Evolution of the critic loss. To yield a fair comparison of methods applying different regularization terms, we display values of the critic’s loss functions without the regularization term throughout training. Results for WGAN-GPs and WGAN-LPs are shown in Figure 4.
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The optimization of the critic with the GP-penalty and $\lambda = 5$ is very unstable: the loss is oscillating heavily around 0. When we use the LP-penalty instead, the critic’s loss smoothly reduces to zero, which is what we expect when the generative distribution $\nu$ steadily converges to the empirical distribution $\mu$ . Also note that we would expect the negative of the critic’s loss to be slightly positive, as a good critic function assigns higher values to real data points $x \sim \mu$ and lower values to generated points $y \sim \nu$ . This is exactly what we observe when using the LP-penalty Interestingly, when using the LP-penalty in combination with a very high penalty weight, like $\lambda = 1 0 0$ , we obtain the same results, indicating that the constraint is always fulfilled for $\lambda = 1 0$ already. Using $\lambda = 1$ in combination with the GP-penalty on the other hand stabilized training but still results in fluctuations in the beginning of the training (results shown in Appendix D.2).
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Figure 3: Level sets of the critic $f$ of WGANs during training, after 10, 50, 100, 500, and 1000 iterations. Yellow corresponds to high, purple to low values of $f$ . Training samples are indicated in red, generated samples in blue. Top: GP-penalty with $\lambda = 1 0$ . Middle: GP-penalty with $\lambda = 1$ . Bottom: LP-penalty with $\lambda = 1 0$ .
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Figure 4: Evolution of the negative of WGAN critic’s loss (without the regularization term) for $\lambda = 5$ . Median results over the 20 runs (blue area indicates quantiles, green dots outliers). Left: For the GP-penalty. Right: For the LP-penalty.
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Estimating the Wasserstein distance. In order to estimate how the actual Wasserstein distance between the real and generated distribution evolves during training, we compute the cost of minimum assignment based on Euclidean distance between sets of samples from the real and generated distributions, using the Kuhn-Munkres algorithm (Kuhn, 1955). We use a sample set size of 500 to maintain reasonable computation time and estimate the distance every 10th iteration over the course of 500 iterations. All experiments were repeated 10 times for different random seeds. From the results for WGAN-GP and WGAN-LP with $\lambda = 5$ shown in Figure 5, we conclude that the proposed LP-penalty leads to smaller estimated Wasserstein distance and less fluctuations during training.
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Table 1: Inception Score on CIFAR-10. Reported are the maximal mean values reached during training. Means are computed over 10 image sets, variances given in parenthesis.
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<table><tr><td>PENALTYWEIGHT</td><td>WGAN-GP</td><td>WGAN-LP</td></tr><tr><td></td><td></td><td></td></tr><tr><td>0.1</td><td>7.781 (± 0.104)</td><td>8.017(± 0.075)</td></tr><tr><td>5</td><td>7.817 (± 0.095)</td><td>7.859 (± 0.085)</td></tr><tr><td>10</td><td>7.840 (± 0.066)</td><td>7.989 (± 0.119)</td></tr><tr><td>100</td><td>7.548 (± 0.102)</td><td>7.815 (± 0.038)</td></tr><tr><td>200</td><td>7.472 (± 0.070)</td><td>7.721 (± 0.105)</td></tr></table>
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When training WGAN-GPs with a regularization parameter of $\lambda = 1$ , training is stabilized as well (see Appendix D.3), indicating that the effect of using a GP-penalty is highly dependent on the right choice of $\lambda$ .
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Figure 5: Evolution of the approximated Wasserstein-1 distance during training of WGANs $\lambda = 5$ , median results over 10 runs). Left: For the GP-penalty. Right: For the LP-penalty.
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Sample quality on CIFAR-10. We trained WGANs with the same ResNet generator and discriminator and the same hyperparameters as Gulrajani et al. (2017) and computed the Inception score (Salimans et al., 2016) throughout training (plots can be found in Appendix D.6). The maximal scores reached in 100000 training iterations with different regularization parameters are reported in Table 1. WGAN-LP reaches the similar or slightly better Inception score as WGAN-GP with small penalty weight $( \lambda \leq 1 0 )$ , while being more stable to other choices of this hyperparameter. This is especially interesting in the light of a recent large scale study, which also reported a strong dependence of sample quality on $\lambda$ for WGAN-GP (see, Figure 8 and 9 in Lucic et al., 2017). Another interesting observation can be made by monitoring the value of the regularization term during training, as in Figure 6), where contributions to the penalty from $| | \nabla f ( \hat { x } ) \bar { | | } > 1$ are shown in the upper and contributions $| | \nabla f ( \hat { x } ) | | < 1$ (only existing for WGAN-GP) are shown in the lower half plane. While the values of the one-sided regularization of WGAN-LP are only slightly larger for larger $\lambda$ (100 compared to 5) the regularization of WGAN-GP shows a strong dependence on the choice of the regularization parameter. For $\lambda = 5$ the penalty contributions from gradient norms smaller than one almost vanished (we found this getting even more severe for even smaller regularization parameters). That is, in a setting where WGAN-GP is performing fine it actually acts similar to WGAN-LP.
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Related penalties We tested the effects of using the regularization terms given by Equation (7) and Equation (9) instead of the the proposed regularization given in Equation (8). Both lead to good performance on toy data but to considerably worse results on CIFAR-10, where training was very unstable. Results are shown in Appendix D.7
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Figure 6: Comparison of the magnitude of the gradient penalty during training on CIFAR, showing $< 1$ and $> 1$ contributions (i.e. $\mathbf { \dot { m } } \mathrm { i n } ( 0 , | | \nabla f ( \mathbf { \dot { \hat { x } } } ) | | - 1 ) ^ { \mathbf { \dot { 2 } } }$ resp. $\operatorname* { m a x } ( \bar { 0 } , | | \nabla f ( \bar { { \boldsymbol x } } ) | | - 1 ) ^ { 2 } )$ ). Left: for regularization parameter $\lambda = 5$ . Right: for regularization parameter $\lambda = 1 0 0$ . The (one-sided) gradient penalty of WGAN-LP is depicted in blue (solid), the gradient penalty of WGAN-GP in red (dashed). All the values for every iteration (one mini-batch) are shown in light blue and red. Dark blue and red lines show the mean over a sliding window of size 500. The figure shows that the part of the gradient penalty of WGAN-GP penalizing a gradient $\leq 1$ almost vanishes for a small regularization parameter, bringing it close to WGAN-LP. For larger values of the regularization parameter, the total penalty of WGAN-GP and its contributing parts are larger than the penalty of WGAN-LP, however, the performance of WGAN-GP suffers more.
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# 7 CONCLUSION
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For stable training of Wasserstein GANs, we propose to use the following penalty term to enforce the Lipschitz constraint that appears in the objective function:
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$$
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\mathbb { E } _ { \hat { x } \sim \tau } [ ( \operatorname* { m a x } \left\{ 0 , \vert \vert \nabla f ( \hat { x } ) \vert \vert - 1 \right\} ) ^ { 2 } ] \tau .
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$$
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We presented theoretical and empirical evidence that this gradient penalty performs better than the previously considered approaches of clipping weights and of applying the stronger gradient penalty given by $\mathbb { E } _ { \hat { x } \sim \tau } [ ( | | \nabla f ( \bar { x } ) | | _ { 2 } - 1 ) ^ { 2 } ]$ . In addition to more stable learning behavior, the proposed regularization term leads to lower sensitivity to the value of the penalty weight $\lambda$ (demonstrating smooth convergence and well-behaved critic scores throughout the whole training process for different values of $\lambda$ ).
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# ACKNOWLEDGMENTS
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This work is supported in part by the European Union under the Horizon 2020 Framework Program for the project WDAqua (GA 642795).
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The authors thank the anonymous reviewers for their valuable suggestions.
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# REFERENCES
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Mart´ın Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial ´ networks. arXiv e-print, arXiv:1701.04862, 2017.
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Mart´ın Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks. ´ In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, pp. 214–223, 2017.
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Marc G. Bellemare, Ivo Danihelka, Will Dabney, Shakir Mohamed, Balaji Lakshminarayanan, Stephan Hoyer, and Remi Munos. The Cramer distance as a solution to biased Wasserstein gra- ´ dients. arXiv e-print, arXiv:1705.10743, 2017.
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Mathieu Blondel, Vivien Seguy, and Antoine Rolet. Smooth and sparse optimal transport. arXiv e-prints, arXiv:1710.06276, 2017.
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Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in Neural Information Processing Systems 29, pp. 2172–2180, 2016.
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Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems 26, pp. 2292–2300, 2013.
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Arnaud Dessein, Nicolas Papadakis, and Jean-Luc Rouas. Regularized optimal transport and the rot mover’s distance. arXiv e-prints, arXiv:1610.06447, 2016.
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David A. Edwards. On the Kantorovich–Rubinstein theorem. Expositiones Mathematicae, 29(4): 387 – 398, 2011.
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Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems 27, pp. 2672–2680, 2014.
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Ishaan Gulrajani, Faruk Ahmed, Mart´ın Arjovsky, Vincent Dumoulin, and Aaron C. Courville. Improved training of Wasserstein GANs. arXiv e-prints, arXiv:1704.00028, 2017.
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Naveen Kodali, Jacob D. Abernethy, James Hays, and Zsolt Kira. How to train your DRAGAN. arXiv e-prints, arXiv:1705.07215v3, 2017.
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Harold W. Kuhn. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2:83–97, 1955.
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Mario Lucic, Karol Kurach, Marcin Michalski, Sylvain Gelly, and Olivier Bousquet. Are GANs created equal? A large-scale study. arXiv e-prints, arXiv:1711.10337, 2017.
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Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems 29, pp. 271–279. 2016.
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Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv e-prints, arXiv:1511.06434, 2015.
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Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems 29, pp. 2234–2242. 2016.
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Gabor J Sz ´ ekely and Maria L Rizzo. Energy statistics: A class of statistics based on distances. ´ Journal of statistical planning and inference, 143(8):1249–1272, 2013.
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Tieleman Tijmen and Geoffrey Hinton. Lecture 6.5—RmsProp: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 2012.
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Ilya Tolstikhin, Sylvain Gelly, Olivier Bousquet, Carl-Johann Simon-Gabriel, and Bernhard Scholkopf. Adagan: Boosting generative models. ¨ arXiv e-prints, arXiv:1701.02386, 2017.
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Cedric Villani. ´ Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2008. ISBN 9783540710509.
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# A PROPERTIES OF AN OPTIMAL CRITIC FUNCTION OF WGANS
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An issue of the original GAN discriminator was that it outputs zero every time it is certain to see generated data, independent on how far away a generated data point lies from the real distribution. As a consequence, locally, there is no incentive for the generator to rather generate a value closer to (but still off) the real data; GAN critic’s optimal value is zero in either case. The WGAN’s optimal critic function measures this distance which helps for the generated distribution to converge, but the interpretation of the absolute value as indicating real (close to 1) and fake data (close to 0) is lost. And worse, there is even no guarantee that the relative values of the optimal critic function help to decide what is real and what is fake. Although this does not seem to cause major problems for the iterative training procedure in practice, we still consider it worthwhile to give a specific example justifying the following observation.
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Observation 4. The WGAN generator could learn wrong things, basing its decision on the values of the optimal critic function, i.e., if it generates at locations of high critic function values.
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Consider the following setting, where the X’s represent generated and the O’s represent real data points.
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Figure 7: Values of the WGAN critic function for some generated data points can be higher than the critic’s values for some real data points. Thus, fake and real points can not be distinguished based on the critics values alone. Real data points are represented by O, generated by X.
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An optimal coupling in this example is quite obvious: We connect the left-most O with the X on the left, and then extend by an arbitrary matching of the other O’s with the other X’s. It is then not hard to verify that the indicated critic function with slope 1 or $- 1$ almost everywhere leads to an equality in the Kantorovich duality and hence is optimal. The value of the critic function at the left-most X is higher than the value at the right-most O, suggesting to generate images at the wrong position. This issue might be fixed by the alternating updates of generator and critic at a later stage of training when less X’s are generated so far on the right side of the O’s. The critic function will then flatten the peak, eventually assigning a lower value to an $\mathrm { X }$ on the left than to any of the O’s.
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Remark 1. The same holds (with only a slight change of the critic function $f$ ) if the $X$ ’s and $O _ { s }$ denote the centers of Gaussians. This can be shown with similar arguments as those in the proofs in Appendix C.3.
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# A.1 PROVING OPTIMALITY OF CERTAIN COMBINATIONS OF COUPLING AND CRITIC FUNCTION
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We show here that the coupling and critic function indicated in Figure 1 are indeed optimal.
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In the one-dimensional example on the left, $\begin{array} { r } { \int _ { \mathbb { R } \times \mathbb { R } } | x - y | d \pi ^ { * } ( x , y ) = \frac { 1 } { 7 } ( 1 + 1 + 1 + 1 + 1 + 1 + 1 ) = } \end{array}$ $\begin{array} { r } { \int _ { \mathbb { R } } f ^ { * } ( x ) d \mu ( x ) - \int _ { \mathbb { R } } f ^ { * } ( x ) d \nu ( x ) } \end{array}$ and thus $\pi ^ { * }$ and $f ^ { * }$ are indeed optimal.
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In the two-dimensional example on the right, the coupling indicated in red and the critic function (described by its function values in blue) are optimal, since with this choice we have
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+
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$\begin{array} { r } { \int _ { \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } } | | x - y | | _ { 2 } \ d \pi ^ { * } ( x , y ) \ = \ \frac { 1 } { 2 } ( 1 + 1 ) \ = \ 1 } \end{array}$ and $\begin{array} { r } { \int _ { \mathbb { R } ^ { n } } f ^ { * } ( y ) \ d \nu ( y ) \ - \ \int _ { \mathbb { R } ^ { n } } f ^ { * } ( x ) \ d \mu ( x ) \ = } \end{array}$ $\textstyle { \frac { 1 } { 2 } } ( 1 + a + 1 ) - { \frac { 1 } { 2 } } ( 0 + a ) = 1$ . Equality of the left hand side and right hand side of the equation proves optimality on both sides.
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+
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# B THE ISSUE WITH THE WEIGHT CLIPPING APPROACH
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The critic function of WGAN is given by a neural network, which raises the question on how to enforce the 1-Lipschitz constraint in the maximization problem of the objective in Equation (5). As Arjovsky et al. (2017) point out, it does not matter whether to maximize over 1-Lipschitz or $\alpha$ - Lipschitz continuous functions, since we can equivalently optimize $\alpha \cdot W ( \mu , \nu )$ instead of $W ( \mu , \nu )$ . An easy consideration leads to the following lemma.
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+
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Lemma 1. The optimal critic function $f ^ { * }$ (leading to the maximum in Eq. (4)) exhausts the Lipschitz constraint for given $\alpha$ in the sense that there is a pair of points $( x , y )$ such that $f ^ { * } ( x ) - f ^ { * } ( y ) =$ $\alpha | | x - y | | _ { 2 }$ .
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| 281 |
+
Proof. If supx6=y n f ∗(y)−f ∗(x)||x−y|| o = c < α , then g = 1c f ∗ generates a contradiction to the optimality of $f ^ { * }$ . (Alternatively, in the case $\alpha = 1$ , it follows directly from Theorem 1, (ii), that the transport is optimal if and only if the Lipschitz constraint of one is exhausted for any two points of the coupling.) □
|
| 282 |
+
|
| 283 |
+
Observation 5. Weight clipping is not a good strategy to enforce the Lipschitz constraint for the critic function.
|
| 284 |
+
|
| 285 |
+
First note that by clipping the weights we enforce a common Lipschitz constraint, where the common Lipschitz constant $\bar { \alpha }$ is defined as the minimal $\alpha \in \mathbb { R }$ such that ${ \overline { { f ( x ) - f ( y ) } } } \leq \alpha \| x - y \| _ { 2 }$ for all $x , y$ and all functions $f$ that can be generated by the network under weight clipping. The actual value of $\bar { \alpha }$ does not follow directly from the weight clipping constant $c _ { \mathrm { m a x } }$ but can be computed from the structure of the network. From Lemma 1 we know that an optimal $f ^ { * }$ exhausts the Lipschitz constraint. We will now show exemplarily for deep NN with rectified linear unit (ReLU) activation functions that there is an extremely limited number of functions generated by the NN using weight clipping that do exhaust the implicitly given common Lipschitz constraint $\bar { \alpha }$ . It follows that, in almost all cases, the optimal $f ^ { * }$ is not in the class of functions that can be generated by the network under the weight clipping constraint.
|
| 286 |
+
|
| 287 |
+
Proposition 4. Consider a (deep) NN with ReLU activation functions and linear output layer. A function generated by the NN under constraining each weight in absolute value by $c _ { m a x }$ exhausts the common Lipschitz constraint if and only if
|
| 288 |
+
|
| 289 |
+
(a) The weight matrix of the first layer consists of constant columns with value $c _ { m a x } o r - c _ { m a x } .$
|
| 290 |
+
|
| 291 |
+
$( b )$ The weights of all other layers are given by a matrix $C ^ { m a x }$ with every entry equal to $c _ { m a x }$
|
| 292 |
+
|
| 293 |
+
Proof. We need to determine every function $f ^ { * }$ generated by the neural network, such that we can find points $x ^ { * } \neq y ^ { * }$ with $f ^ { * } ( y ^ { * } ) - f ^ { * } ( x ^ { * } ) = \bar { \alpha } | | x ^ { * } - y ^ { * } | | _ { 2 }$ . Recall that $\bar { \alpha }$ is defined as the minimal $\alpha$ satisfying $f ( y ) - f ( x ) \leq \alpha \vert \vert x - y \vert \vert _ { 2 }$ for all functions $f$ generated by the neural network and all points $x , y$ .
|
| 294 |
+
|
| 295 |
+
In the following, we will denote by $\alpha ( f )$ the Lipschitz constant of $f$ , i.e., the smallest $\alpha \in \mathbb { R }$ such that $f ( x ) - f ( y ) \leq \alpha \vert \vert x - y \vert \vert _ { 2 }$ for all $x , y$ .
|
| 296 |
+
|
| 297 |
+
Every function generated by the neural net is a composition of functions
|
| 298 |
+
|
| 299 |
+
$$
|
| 300 |
+
f = f _ { n } \circ { \mathrm { r e l u } } \circ f _ { n - 1 } \circ . . . \circ { \mathrm { r e l u } } \circ f _ { 1 } .
|
| 301 |
+
$$
|
| 302 |
+
|
| 303 |
+
with linear functions $f _ { i }$ and relu denoting a layer of activation functions with rectifier linear units. Since each linear function $f _ { i }$ is Lipschitz continuous with Lipschitz constant $\alpha ( f _ { i } )$ and relu is Lipschitz continuous with $\alpha ( \mathrm { r e l u } ) = 1$ , it follows that $f$ is Lipschitz continuous with $\begin{array} { r } { \dot { \alpha } ( f ) \le \prod _ { i } \alpha ( \bar { f } _ { i } ) } \end{array}$ . Moreover, equality holds if there is a pair of points $( x , y )$ such that the consecutive images witness the maximal Lipschitz constant $\alpha ( f _ { i } )$ and $\alpha ( \mathrm { { r e l u } ) }$ for each of the individual functions making up the composition of $f$ . More formally, equality holds if and only if there is a tuple of pairs of points $( \boldsymbol { x } ^ { ( i ) } , \boldsymbol { y } ^ { ( \bar { i } ) } )$ , $1 \leq i \leq n - 1$ , such that for all $1 \leq i \leq n$ ,
|
| 304 |
+
|
| 305 |
+
(i) $\boldsymbol { x } ^ { ( i ) } \neq \boldsymbol { y } ^ { ( i ) }$
|
| 306 |
+
|
| 307 |
+
$$
|
| 308 |
+
| f _ { i } ( x ^ { ( i ) } ) - f _ { i } ( y ^ { ( i ) } ) | = \alpha ( f _ { i } ) | | x ^ { ( i ) } - y ^ { ( i ) } | | _ { 2 }
|
| 309 |
+
$$
|
| 310 |
+
|
| 311 |
+
(iv) All entries of $f _ { i } ( x ^ { ( i ) } )$ and $f _ { i } ( y ^ { ( i ) } )$ are larger or equal to zero. This is equivalent to the condition that
|
| 312 |
+
|
| 313 |
+
$$
|
| 314 |
+
\begin{array} { r } { | \mathrm { r e l u } ( f _ { i } ( x ^ { ( i ) } ) ) - \mathrm { r e l u } ( f _ { i } ( y ^ { ( i ) } ) ) | = \alpha ( \mathrm { r e l u } ) | | f _ { i } ( x ^ { ( i ) } ) - f _ { i } ( y ^ { ( i ) } ) | | _ { 2 } \ . } \end{array}
|
| 315 |
+
$$
|
| 316 |
+
|
| 317 |
+
It follows that to determine $f ^ { * }$ we need to maximize $\alpha ( f _ { i } )$ for the linear layers with weight constraint $c _ { \mathrm { m a x } }$ and find a sequence of points $( x ^ { ( i ) } , y ^ { ( i ) } )$ that satisfy (i)-(iv). The existence of the sequence of points shows that $\begin{array} { r } { \bar { \alpha } ( f ^ { * } ) = \bar { \prod } _ { i = 1 } ^ { n } \alpha ( f _ { i } ) } \end{array}$ and maximizing each $\alpha ( f _ { i } )$ then shows that
|
| 318 |
+
|
| 319 |
+
$$
|
| 320 |
+
\alpha ( f ^ { * } ) = \prod _ { i = 1 } ^ { n } \alpha ( f _ { i } ) = \bar { \alpha } .
|
| 321 |
+
$$
|
| 322 |
+
|
| 323 |
+
Since, as we will show, the conditions in (a) and (b) maximize the Lipschitz constraint of each layer individually, the existence of suitable $( x ^ { ( i ) } , y ^ { ( i ) } )$ proves the if-direction of the proposition.
|
| 324 |
+
|
| 325 |
+
For the only-if direction, we will see that the ability to find the sequence of points gives restrictions on how to maximize $\alpha ( f _ { i } )$ of an individual layer, leading to the more restrictive condition of (b) for all but the first layer (cf. (a)).
|
| 326 |
+
|
| 327 |
+
So let us first maximize the Lipschitz constraint of each linear layer and then make sure that we can find the corresponding points. We write the linear layer as a matrix multiplication $f _ { i } ( x ) = A ^ { ( i ) } x$ . Using linearity,
|
| 328 |
+
|
| 329 |
+
$$
|
| 330 |
+
\alpha ( f _ { i } ) = \operatorname* { m a x } _ { | | z | | _ { 2 } = 1 } | | A ^ { ( i ) } z | | _ { 2 } \ ,
|
| 331 |
+
$$
|
| 332 |
+
|
| 333 |
+
and our goal can be reformulated to finding the matrix $A ^ { ( i ) }$ maximizing $\alpha ( f _ { i } )$
|
| 334 |
+
|
| 335 |
+
For any fixed $z$ , $| | A ^ { ( i ) } z | | _ { 2 }$ is maximized exactly when each vector entry is maximized in absolute value. Now, with $A ^ { ( i ) } = ( a _ { j , k } ^ { ( i ) } ) _ { j , k }$ and $\operatorname { s g n } ( { \mathord { \cdot } } )$ denoting the sign function,
|
| 336 |
+
|
| 337 |
+
$$
|
| 338 |
+
| ( A ^ { ( i ) } z ) _ { j } | = \left| \sum _ { k } a _ { j , k } ^ { ( i ) } z _ { k } \right| \leq \sum _ { k } | a _ { j , k } ^ { ( i ) } | | z _ { k } | \leq \sum _ { k } c _ { \operatorname* { m a x } } | z _ { k } | \mathrm { ( b y ~ t h e ~ w e i g h t ~ c o n s t r a i n t ) }
|
| 339 |
+
$$
|
| 340 |
+
|
| 341 |
+
$$
|
| 342 |
+
= \sum _ { k } ( c _ { \mathrm { m a x } } \cdot \mathrm { s g n } ( z _ { k } ) ) \cdot z _ { k }
|
| 343 |
+
$$
|
| 344 |
+
|
| 345 |
+
and equality holds if and only if $A ^ { ( i ) }$ or $- A ^ { ( i ) }$ consists of columns of constant entry with the value $c _ { \mathrm { m a x } } \cdot \mathrm { s g n } ( z _ { k } )$ in column $k$ . It follows that
|
| 346 |
+
|
| 347 |
+
$$
|
| 348 |
+
\alpha ( f _ { i } ) = \operatorname* { m a x } _ { | | z | | _ { 2 } = 1 } | | A ^ { ( i ) } z | | _ { 2 } = \operatorname* { m a x } _ { | | z | | _ { 2 } = 1 } c _ { \operatorname* { m a x } } \cdot | | z | | _ { 1 }
|
| 349 |
+
$$
|
| 350 |
+
|
| 351 |
+
$$
|
| 352 |
+
= c _ { \operatorname* { m a x } } \sqrt { d i m ( z ) }
|
| 353 |
+
$$
|
| 354 |
+
|
| 355 |
+
with equality if and only if $\begin{array} { r } { z _ { k } = \pm \frac { 1 } { \sqrt { d i m ( z ) } } } \end{array}$ for all $k$ .
|
| 356 |
+
|
| 357 |
+
Hence, for the first linear layer we need to choose a matrix $A ^ { ( 1 ) }$ satisfying (a) of the statement of the proposition.
|
| 358 |
+
|
| 359 |
+
Now, we find a pair $( x ^ { ( 1 ) } , y ^ { ( 1 ) } )$ with
|
| 360 |
+
|
| 361 |
+
$$
|
| 362 |
+
x ^ { ( 1 ) } - y ^ { ( 1 ) } = a \cdot ( \pm 1 , \pm 1 , \ldots , \pm 1 ) { \mathrm { f o r ~ s o m e ~ } } a \neq 0
|
| 363 |
+
$$
|
| 364 |
+
|
| 365 |
+
such that
|
| 366 |
+
|
| 367 |
+
$$
|
| 368 |
+
\mathtt { s g n } ( x _ { k } ^ { ( 1 ) } ) = \mathtt { s g n } ( y _ { k } ^ { ( 1 ) } ) = \mathtt { s g n } ( x _ { k } ^ { ( 1 ) } - y _ { k } ^ { ( 1 ) } ) = \mathtt { t h e ~ s i g n ~ o f ~ c o l u m n } k \mathrm { ~ o f ~ } A ^ { ( 1 ) } .
|
| 369 |
+
$$
|
| 370 |
+
|
| 371 |
+
This is the only possibility to ensure (iii) and (iv) of the conditions above. Note that also (i) holds for $( x ^ { ( 1 ) } , y ^ { ( 1 ) } )$ , and (ii) (together with (iv)) determines $( x ^ { ( 2 ) } , y ^ { ( 2 ) } )$ uniquely from $( x ^ { ( 1 ) } , y ^ { ( 1 ) } )$ as
|
| 372 |
+
|
| 373 |
+
$$
|
| 374 |
+
\begin{array} { r } { x ^ { ( 2 ) } = A ^ { ( 1 ) } x ^ { ( 1 ) } = c _ { \operatorname* { m a x } } \cdot \vert \vert x ^ { ( 1 ) } \vert \vert _ { 1 } \cdot ( 1 , 1 , . . . , 1 ) } \\ { \& } \end{array} .
|
| 375 |
+
$$
|
| 376 |
+
|
| 377 |
+
We may assume that $| | x ^ { ( 1 ) } | | _ { 1 } ~ > ~ | | y ^ { ( 1 ) } | | _ { 1 }$ . (Otherwise, switch the roles of $x$ and $y$ . In the case of equality, we need to choose a different pair for $( x ^ { ( 1 ) } , y ^ { ( 1 ) } )$ not to violate (i) for $( x ^ { ( 2 ) } , y ^ { ( 2 ) } )$ .) Then we have that $x ^ { ( 2 ) } \neq y ^ { ( 2 ) }$ ,
|
| 378 |
+
|
| 379 |
+
$$
|
| 380 |
+
+ 1 = \mathrm { s g n } ( x _ { k } ^ { ( 2 ) } ) = \mathrm { s g n } ( y _ { k } ^ { ( 2 ) } ) = \mathrm { s g n } ( x _ { k } ^ { ( 2 ) } - y _ { k } ^ { ( 2 ) } ) \mathrm { f o r } \mathrm { a l l } k .
|
| 381 |
+
$$
|
| 382 |
+
|
| 383 |
+
Using the same arguments as above, it follows that for such $( x ^ { ( 2 ) } , y ^ { ( 2 ) } )$ , to maximize the Lipschitz constant of $f _ { 2 }$ (and to guarantee that the maximum is reached at $( x ^ { ( 2 ) } , y ^ { ( 2 ) } ) )$ , we need to have $A ^ { ( 2 ) }$ equal to a matrix with $c _ { \mathrm { m a x } }$ at each position.
|
| 384 |
+
|
| 385 |
+
Now (i)-(iv) also hold for the second layer and one may now proceed by induction to show that for $i \geq 2$ , $A ^ { ( i ) }$ contains only $c _ { \mathrm { m a x } }$ for each of its entries. This is the only way to maximize the Lipschitz constraint for functions generated by the neural net, and it does indeed hold $| | f ^ { * } ( x ^ { * } ) - f ^ { * } ( y ^ { * } ) | | _ { 2 } =$ ${ \bar { \alpha } } | | x - y | | _ { 2 }$ with $x ^ { * } = x ^ { ( 1 ) } , y ^ { * } = y ^ { ( 1 ) }$ and
|
| 386 |
+
|
| 387 |
+
$$
|
| 388 |
+
( x ^ { ( i ) } , y ^ { ( i ) } ) = ( f _ { i } \circ \mathsf { r e l u } \circ f _ { i - 1 } \circ \dots \circ \mathsf { r e l u } \circ f _ { 1 } ( x ^ { * } ) , f _ { i } \circ \mathsf { r e l u } \circ f _ { i - 1 } \circ \dots \circ \mathsf { r e l u } \circ f _ { 1 } ( y ^ { * } ) ) .
|
| 389 |
+
$$
|
| 390 |
+
|
| 391 |
+
# C PROOFS
|
| 392 |
+
|
| 393 |
+
# C.1 PROOF OF THEOREM 1
|
| 394 |
+
|
| 395 |
+
Proof. We provide the arguments how to derive our version from Theorem 5.10 of Villani (2008).
|
| 396 |
+
|
| 397 |
+
With $c ( x , y ) = | | x - y | | _ { 2 }$ , our assumptions imply (with $c _ { \mathcal { X } } = c _ { \mathcal { Y } } = | | \cdot | | _ { 2 } )$ that all conclusions of Theorem 5. $1 0 \ ( i ) - ( i i i )$ hold. Moreover, 5.4 of Villani (2008) shows that in this case $\psi = \psi ^ { c }$ (in the notation of Villani (2008)) and $c$ -convexity is the same as 1-Lipschitz continuity. This leads to our formulation in (i) and the existence of an optimal coupling $\pi ^ { * }$ and an optimal critic function $f ^ { * }$ by part (iii).
|
| 398 |
+
|
| 399 |
+
If we let
|
| 400 |
+
|
| 401 |
+
$$
|
| 402 |
+
\Gamma _ { f } = \{ ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } \mid f ( x ) - f ( y ) = | | x - y | | _ { 2 } \}
|
| 403 |
+
$$
|
| 404 |
+
|
| 405 |
+
then it follows from the proof of Theorem 5.10 that the set $\Gamma$ in part 5.10 (iii) is given by $\Gamma =$ Tf∗∈Lip optimal Γf∗ , where f ∗ being optimal means that it leads to a maximum on the RHS of equation (1).
|
| 406 |
+
|
| 407 |
+
To prove our part $( i i )$ from 5.10, let $\pi ^ { * }$ be optimal. Then, by 5.10 (iii), $\pi ^ { * } ( \Gamma ) = 1$ . Hence, in particular, $\pi ^ { * } ( \Gamma _ { f ^ { * } } ) = 1$ for all optimal $f ^ { * } \in \mathcal { L } i p _ { 1 }$ . This shows that (a) implies (b). For the other direction, we use that if $\pi ^ { * } ( \Gamma _ { f ^ { * } } ) = 1$ for all optimal $f ^ { * }$ , then $\pi ^ { * } ( \Gamma ) = 1$ , which by Theorem 5.10 (iii) is equivalent to $\pi ^ { * }$ being optimal. □
|
| 408 |
+
|
| 409 |
+
# C.2 PROOF OF PROPOSITION 1
|
| 410 |
+
|
| 411 |
+
Proof. It follows from Theorem 1 (ii) that for all $( x , y )$ in the support of $\pi ^ { * }$ we have $\left| f ^ { * } ( y ) - \right.$ $f ^ { * } ( { \dot { x } } ) | = | | x - y | | _ { 2 }$ . Considering the line between $x$ and $y$ , the 1-Lipschitz constraint implies that the values of $f ^ { * }$ have to follow a linear function (since assuming that the slope was smaller than one at some point would imply that the differentiable function must have a slope larger than one somewhere else between $x$ and $y$ , which contradicts the 1-Lipschitz constraint). It follows that at each point on the line, the partial derivative has norm equal to one into the direction pointing from the real data point $x$ to the generated one $y$ (which are coupled by the corresponding optimal coupling). Since, by the 1-Lipschitz constraint, the maximal norm of a partial derivative at any point into any direction is one, the given direction is the direction of maximal descent, i.e. equals the gradient.
|
| 412 |
+
|
| 413 |
+
# C.3 PROOF OF PROPOSITION 2
|
| 414 |
+
|
| 415 |
+
To prove Proposition 2, we first prove that $\phi ^ { * } ( x ) = - | x |$ is the optimal critic function for certain distributions with non-overlapping support, and then reduce the example with Gaussian functions to this simplified setting.
|
| 416 |
+
|
| 417 |
+
Proposition 5. Let $f$ and $g$ be two continuous functions on the real line that satisfy the following conditions:
|
| 418 |
+
|
| 419 |
+
• $f$ and $g$ are symmetric with respect to the y-axis.
|
| 420 |
+
• $f ( x ) \geq 0$ and $g ( x ) \geq 0$ for all $x$ .
|
| 421 |
+
• If $s u p p _ { \circ } ( h ) = \{ x \in \mathbb { R } \mid h ( x ) > 0 \}$ denotes the open support of a continuous function $h$ , then $s u p p _ { \circ } ( f ) \cap s u p p _ { \circ } ( g ) = \emptyset$ .
|
| 422 |
+
• $f$ has connected support (this implies that $f$ is centered around 0 because of the symmetry). • $\begin{array} { r } { \int _ { \mathbb { R } } f ( x ) d x = \int _ { \mathbb { R } } g ( x ) d x . } \end{array}$
|
| 423 |
+
|
| 424 |
+
Then the maximum of $\begin{array} { r } { \int _ { \mathbb { R } } \phi ( x ) ( f ( x ) - g ( x ) ) d x } \end{array}$ over $\phi \in \mathcal { L } i p _ { 1 }$ is maximized for $\phi ^ { * } ( x ) = - | x |$ .
|
| 425 |
+
|
| 426 |
+
Proof. Before going into the technical details, we wish to point out the simple idea of the proof, which is to transport the left/right half of the distribution given by $g$ to the left/right half of the distribution given by $f$ respectively.
|
| 427 |
+
|
| 428 |
+
We first multiply both $f$ and $g$ by a constant number $c$ such that
|
| 429 |
+
|
| 430 |
+
$$
|
| 431 |
+
\int _ { \mathbb { R } } c \cdot f ( x ) d x = \int _ { \mathbb { R } } c \cdot g ( x ) d x = 1 .
|
| 432 |
+
$$
|
| 433 |
+
|
| 434 |
+
Then $c \cdot f$ and $c \cdot g$ define probability density functions. A function $\phi \in \mathcal { L } i p _ { 1 }$ maximizes $\begin{array} { r } { \int _ { \mathbb { R } } \phi ( x ) ( c \cdot } \end{array}$ $f ( x ) - c \cdot g ( x ) ) d x$ if and only if it maximizes $\begin{array} { r } { \int _ { \mathbb { R } } \phi ( x ) ( f ( x ) - g ( x ) ) d x } \end{array}$ . We therefore may assume from now on that
|
| 435 |
+
|
| 436 |
+
$$
|
| 437 |
+
\int _ { \mathbb { R } } f ( x ) d x = \int _ { \mathbb { R } } g ( x ) d x = 1 .
|
| 438 |
+
$$
|
| 439 |
+
|
| 440 |
+
Now it suffices to find a coupling $\pi$ of the probability distributions defined by $f$ and $g$ (that is itself defined by a probability density function $\pi : \mathbb { R } \times \mathbb { R } \to \mathbb { R }$ ) such that for $\phi ( x ) = - | x |$ we get
|
| 441 |
+
|
| 442 |
+
$$
|
| 443 |
+
\int _ { \mathbb { R } \times \mathbb { R } } | x - y | \cdot \pi ( x , y ) d x d y = \int _ { \mathbb { R } } \phi ( x ) ( f ( x ) - g ( x ) ) d x .
|
| 444 |
+
$$
|
| 445 |
+
|
| 446 |
+
The proof then follows from the Kantorovich duality theorem 1, because the right hand side is always smaller or equal to the left hand side for arbitrary coupling $\pi$ and function $\phi \in \mathcal { L } i p _ { 1 }$ and is consequently maximized when equality holds. By the assumption of symmetry, we may write $g = g _ { 1 } + g _ { 2 }$ where the support $\operatorname { s u p p } ( g _ { 1 } ) { \overset { \cdot } { \subseteq } } \{ x \mid x { \overset { \cdot } { < } } 0 \}$ and $g _ { 2 } \bar { ( } x ) = g _ { 1 } \bar { ( } - x )$ for all $x$ . The area under $g _ { 1 } ( x )$ equals half the area under $f ( x )$ , or put differently,
|
| 447 |
+
|
| 448 |
+
$$
|
| 449 |
+
\int _ { \mathbb { R } } g _ { 1 } ( x ) d x = \int _ { \mathbb { R } } f ( x ) \delta _ { ( - \infty , 0 ] } ( x ) d x = \frac { 1 } { 2 } .
|
| 450 |
+
$$
|
| 451 |
+
|
| 452 |
+
We now consider the probability density function $\pi _ { 1 } : \mathbb { R } \times \mathbb { R } \to \mathbb { R }$ given by
|
| 453 |
+
|
| 454 |
+
$$
|
| 455 |
+
\pi _ { 1 } ( x , y ) = 2 g _ { 1 } ( x ) \cdot 2 f ( y ) \cdot \delta _ { ( - \infty , 0 ] } ( y ) ,
|
| 456 |
+
$$
|
| 457 |
+
|
| 458 |
+
which defines a coupling between the two distributions given by the probability density functions $2 g _ { 1 }$ and $2 f \cdot \delta _ { ( - \infty , 0 ] }$ . For later use we note that
|
| 459 |
+
|
| 460 |
+
$$
|
| 461 |
+
\int _ { x \in \mathbb { R } } \pi _ { 1 } ( x , y ) d x = 2 \cdot f ( y ) \cdot \delta _ { ( - \infty , 0 ] } ( y ) { \mathrm { ~ a n d ~ } } \int _ { y \in \mathbb { R } } \pi _ { 1 } ( x , y ) d y = 2 \cdot g _ { 1 } ( x ) .
|
| 462 |
+
$$
|
| 463 |
+
|
| 464 |
+
We define $\pi _ { 2 } ( x , y ) = \pi _ { 1 } ( - x , - y )$ for $y \ne 0$ and $\pi _ { 2 } ( x , 0 ) = 0$ . Further, we let $\pi = { \textstyle { \frac { 1 } { 2 } } } \pi _ { 1 } + { \textstyle { \frac { 1 } { 2 } } } \pi _ { 2 }$ Then $\pi$ defines a coupling between $g$ and $f$ as can be seen by computing
|
| 465 |
+
|
| 466 |
+
$$
|
| 467 |
+
\int _ { x \in \mathbb { R } } \pi ( x , y ) d x = \frac { 1 } { 2 } \int _ { x \in \mathbb { R } } \pi _ { 1 } ( x , y ) d x + \frac { 1 } { 2 } \int _ { x \in \mathbb { R } } \pi _ { 2 } ( x , y ) d x
|
| 468 |
+
$$
|
| 469 |
+
|
| 470 |
+
$$
|
| 471 |
+
= \frac { 1 } { 2 } \int _ { x \in \mathbb { R } } \pi _ { 1 } ( x , y ) d x + \frac { 1 } { 2 } \int _ { x \in \mathbb { R } } \pi _ { 1 } ( - x , - y ) \delta _ { \{ y \neq 0 \} } ( y ) d x
|
| 472 |
+
$$
|
| 473 |
+
|
| 474 |
+
$$
|
| 475 |
+
= f ( y ) \delta _ { ( - \infty , 0 ] } ( y ) + f ( y ) \delta _ { ( 0 , \infty ) } ( y ) = f ( y )
|
| 476 |
+
$$
|
| 477 |
+
|
| 478 |
+
and
|
| 479 |
+
|
| 480 |
+
$$
|
| 481 |
+
\int _ { y \in \mathbb { R } } \pi ( x , y ) d y = \frac { 1 } { 2 } \int _ { y \in \mathbb { R } } \pi _ { 1 } ( x , y ) d y + \frac { 1 } { 2 } \int _ { y \in \mathbb { R } } \pi _ { 2 } ( x , y ) d y
|
| 482 |
+
$$
|
| 483 |
+
|
| 484 |
+
$$
|
| 485 |
+
\frac { 1 } { 2 } \int _ { y \in \mathbb { R } } { \pi } _ { 1 } ( x , y ) d y + \frac { 1 } { 2 } \int _ { y \in \mathbb { R } } { \pi } _ { 1 } ( - x , - y ) \delta _ { \{ y \neq 0 \} } ( y ) d y
|
| 486 |
+
$$
|
| 487 |
+
|
| 488 |
+
$$
|
| 489 |
+
= g _ { 1 } ( x ) + g _ { 1 } ( - x ) = g _ { 1 } ( x ) + g _ { 2 } ( x ) = g ( x )
|
| 490 |
+
$$
|
| 491 |
+
|
| 492 |
+
We have established the existence of some coupling between $f$ and $g$ and we will now compute its transport costs. We will subsequently show that this equals $\begin{array} { r } { \int _ { \mathbb { R } } ( - | x | ) ( f ( x ) - g ( x ) ) d x } \end{array}$ , hence both $\pi$ and $\phi$ are optimal by realizing the Kantorovich duality.
|
| 493 |
+
|
| 494 |
+
We aim at showing $\begin{array} { r } { \int _ { \mathbb { R } \times \mathbb { R } } | x - y | \pi ( x , y ) d x d y = \int _ { \mathbb { R } } ( - | x | ) ( f ( x ) - g ( x ) ) d x . } \end{array}$
|
| 495 |
+
|
| 496 |
+
$$
|
| 497 |
+
\int _ { \mathbb { R } \times \mathbb { R } } | x - y | \pi ( x , y ) d x d y \overset { s y m m e t r y } { = } \int _ { \mathbb { R } \times \mathbb { R } } | x - y | \pi _ { 1 } ( x , y ) d x d y
|
| 498 |
+
$$
|
| 499 |
+
|
| 500 |
+
$$
|
| 501 |
+
= \int _ { \mathbb { R } \times \mathbb { R } } ( y - x ) \pi _ { 1 } ( x , y ) d x d y .
|
| 502 |
+
$$
|
| 503 |
+
|
| 504 |
+
The latter equation holds because for $( x , y )$ in the support of $\pi _ { 1 }$ we have $x \leq y$ . (To see this, note that support of $\pi _ { 1 }$ is a subset of the support of $g _ { 1 } \times \left( f \cdot \delta _ { ( - \infty , 0 ] } \right) .$ .) Let
|
| 505 |
+
|
| 506 |
+
$$
|
| 507 |
+
x _ { 0 } = \frac { \int _ { \mathbb { R } } x \cdot g _ { 1 } ( x ) d x } { \int _ { \mathbb { R } } g _ { 1 } ( x ) d x } , \mathrm { ~ a n d ~ } y _ { 0 } = \frac { \int _ { \mathbb { R } } y \cdot f ( y ) \cdot \delta _ { ( - \infty , 0 ) } ( y ) d y } { \int _ { \mathbb { R } } f ( y ) \cdot \delta _ { ( - \infty , 0 ) } ( y ) d y } .
|
| 508 |
+
$$
|
| 509 |
+
|
| 510 |
+
Then
|
| 511 |
+
|
| 512 |
+
$$
|
| 513 |
+
\int _ { \mathbb { R } } ( x - x _ { 0 } ) \cdot g _ { 1 } ( x ) d x = 0 { \mathrm { ~ a n d ~ } } \int _ { \mathbb { R } } ( y - y _ { 0 } ) \cdot f ( y ) \cdot \delta _ { ( - \infty , 0 ] } ( y ) d y = 0 .
|
| 514 |
+
$$
|
| 515 |
+
|
| 516 |
+
Now, it follows that
|
| 517 |
+
|
| 518 |
+
$$
|
| 519 |
+
\int _ { \mathbb { R } \times \mathbb { R } } ( y - x ) \pi _ { 1 } ( x , y ) d x d y = \int _ { \mathbb { R } \times \mathbb { R } } ( y - y _ { 0 } + y _ { 0 } - x ) \pi _ { 1 } ( x , y ) d x d y
|
| 520 |
+
$$
|
| 521 |
+
|
| 522 |
+
$$
|
| 523 |
+
= \int _ { x } \int _ { y } ( y - y _ { 0 } ) \pi _ { 1 } ( x , y ) d x d y + \int _ { x } \int _ { y } ( y _ { 0 } - x ) \pi _ { 1 } ( x , y ) d x d y
|
| 524 |
+
$$
|
| 525 |
+
|
| 526 |
+
$$
|
| 527 |
+
= \int _ { y } ( y - y _ { 0 } ) \int _ { x } \pi _ { 1 } ( x , y ) d x d y + \int _ { x } \int _ { y } ( y _ { 0 } - x ) \pi _ { 1 } ( x , y ) d x d y
|
| 528 |
+
$$
|
| 529 |
+
|
| 530 |
+
$$
|
| 531 |
+
= 2 \underbrace { \int _ { y } ( y - y _ { 0 } ) \cdot f ( y ) \cdot \delta _ { ( - \infty , 0 ] } ( y ) d y } _ { = 0 } + \int _ { x } \int _ { y } ( y _ { 0 } - x _ { 0 } + x _ { 0 } - x ) \pi _ { 1 } ( x , y ) d x d y
|
| 532 |
+
$$
|
| 533 |
+
|
| 534 |
+
$$
|
| 535 |
+
= ( y _ { 0 } - x _ { 0 } ) \int _ { x } \int _ { y } \pi _ { 1 } ( x , y ) d x d y + \int _ { x } ( x _ { 0 } - x ) \underbrace { \int _ { y } \pi _ { 1 } ( x , y ) d y } _ { = 2 g _ { 1 } ( x ) } d x
|
| 536 |
+
$$
|
| 537 |
+
|
| 538 |
+
Hence,
|
| 539 |
+
|
| 540 |
+
$$
|
| 541 |
+
\int _ { { \mathbb R } \times { \mathbb R } } | x - y | \pi ( x , y ) d x d y = ( y _ { 0 } - x _ { 0 } ) = \frac { \int _ { { \mathbb R } } y \cdot f ( y ) \cdot \delta _ { ( - \infty , 0 ) } ( y ) d y } { \frac { 1 } { 2 } } - \frac { \int _ { { \mathbb R } } x \cdot g _ { 1 } ( x ) d x } { \frac { 1 } { 2 } } .
|
| 542 |
+
$$
|
| 543 |
+
|
| 544 |
+
$$
|
| 545 |
+
\begin{array} { l } { { \displaystyle = 2 \int _ { \mathbb R } x \cdot ( f ( x ) \cdot \delta _ { ( - \infty , 0 ) } ( x ) - g _ { 1 } ( x ) ) d x } } \\ { { \displaystyle \quad = 2 \int _ { - \infty } ^ { 0 } x \cdot ( f ( x ) - g ( x ) ) d x } } \\ { { \displaystyle \quad = 2 \int _ { - \infty } ^ { 0 } ( - | x | ) \cdot ( f ( x ) - g ( x ) ) d x } } \\ { { \displaystyle \quad \it { s y m m e t r y } \int _ { \mathbb R } ( - | x | ) \cdot ( f ( x ) - g ( x ) ) d x } } \end{array}
|
| 546 |
+
$$
|
| 547 |
+
|
| 548 |
+
We are now able to proof Proposition 2
|
| 549 |
+
|
| 550 |
+
Proof to Proposition 2. Let $f$ denote the probability density function of $\mathcal { N } ( 0 , 1 )$ and $\begin{array} { r } { g = \frac { 1 } { 2 } g _ { - 1 } + } \end{array}$ $\textstyle { \frac { 1 } { 2 } } g _ { 1 }$ denote the sum of half the probability density functions $g _ { - 1 }$ of $\mathcal { N } ( - 1 , 1 )$ and $g _ { 1 }$ of $\mathcal { N } ( 1 , 1 )$ . Let
|
| 551 |
+
|
| 552 |
+
$$
|
| 553 |
+
\tilde { f } ( x ) = \operatorname* { m a x } \left\{ 0 , \left( f ( x ) - g ( x ) \right) \right\} \mathrm { ~ a n d ~ } \tilde { g } ( x ) = \operatorname* { m a x } \left\{ 0 , \left( g ( x ) - f ( x ) \right) \right\} ,
|
| 554 |
+
$$
|
| 555 |
+
|
| 556 |
+
i.e. $\tilde { f }$ and $\tilde { g }$ are the positive and the negative part of $( f - g )$ . Then $\tilde { f }$ and $\tilde { g }$ satisfy the hypothesis of Proposition 5 and the maximum
|
| 557 |
+
|
| 558 |
+
$$
|
| 559 |
+
\operatorname* { m a x } _ { \phi \in \mathcal { L } i p _ { 1 } } \int _ { \mathbb { R } } \phi ( x ) ( f ( x ) - g ( x ) ) d x = \operatorname* { m a x } _ { \phi \in \mathcal { L } i p _ { 1 } } \int _ { \mathbb { R } } \phi ( x ) ( \tilde { f } ( x ) - \tilde { g } ( x ) ) d x
|
| 560 |
+
$$
|
| 561 |
+
|
| 562 |
+
is obtained for $\phi ^ { * } ( x ) = - | x |$ .
|
| 563 |
+
|
| 564 |
+
# C.4 PROOF OF PROPOSITION 3
|
| 565 |
+
|
| 566 |
+
Proof. For any fixed function $f$ and $\lambda > 0$ , the two regularized losses of the critic function $f$ are of the form
|
| 567 |
+
|
| 568 |
+
$$
|
| 569 |
+
\mathcal { L } _ { \lambda } ^ { L P } ( f ) = c + \lambda \int \operatorname* { m a x } \{ 0 , ( h ( z ) - 1 ) \} ^ { 2 } ) \mathrm { d } \tau ( z ) \mathrm { a n d } \mathcal { L } _ { \lambda } ^ { G P } ( f ) = c + \lambda \int ( h ( z ) - 1 ) ^ { 2 } ) \mathrm { d } \tau ( z )
|
| 570 |
+
$$
|
| 571 |
+
|
| 572 |
+
for some real number $c$ , a function $h$ with with $h ( z ) \geq 0$ for all $z$ and a probability distribution $\tau$ Since for any real number $0 \leq a$ we have that
|
| 573 |
+
|
| 574 |
+
$$
|
| 575 |
+
\operatorname* { m a x } \{ 0 , ( a - 1 ) \} ^ { 2 } \leq ( a - 1 ) ^ { 2 } \leq \operatorname* { m a x } \{ 0 , ( a - 1 ) \} ^ { 2 } + 1
|
| 576 |
+
$$
|
| 577 |
+
|
| 578 |
+
it follows that
|
| 579 |
+
|
| 580 |
+
$$
|
| 581 |
+
\begin{array} { r } { \mathcal { L } _ { \lambda } ^ { L P } ( f ) \leq \mathcal { L } _ { \lambda } ^ { G P } ( f ) \leq \mathcal { L } _ { \lambda } ^ { L P } ( f ) + \lambda . } \end{array}
|
| 582 |
+
$$
|
| 583 |
+
|
| 584 |
+
Therefore the inequalities also hold for the infimum over a class of functions, hence
|
| 585 |
+
|
| 586 |
+
$$
|
| 587 |
+
\begin{array} { r } { \mathcal { L } _ { \lambda } ^ { L P } \leq \mathcal { L } _ { \lambda } ^ { G P } \leq \mathcal { L } _ { \lambda } ^ { L P } + \lambda . } \end{array}
|
| 588 |
+
$$
|
| 589 |
+
|
| 590 |
+
# C.5 THE ARGUMENTS SUPPORTING OBSERVATION 3
|
| 591 |
+
|
| 592 |
+
For the coupled pairs $( x , y )$ and $( x , y ^ { \prime } )$ we have that the partial derivatives at $x$ into the directions of $y$ and $y ^ { \prime }$ respectively have an absolute value of one. If there are two such directions and $f ^ { * }$ is differentiable, then the norm of its gradient must be larger than one, contradicting the 1-Lipschitz constraint. Indeed, recall that, considering $f$ as a function on the line $\{ x + \lambda \cdot v \mid \lambda \in \mathbb { R } \}$ with $v$ of unit length, the slope of $f$ at $x$ is given by $\nabla f ( x ) \cdot v = D _ { v } ( f ( x ) )$ . Now
|
| 593 |
+
|
| 594 |
+
$$
|
| 595 |
+
\nabla f ( \boldsymbol { x } ) \cdot \boldsymbol { v } = | | \nabla f ( \boldsymbol { x } ) | | _ { 2 } \cdot \cos ( \theta _ { v } )
|
| 596 |
+
$$
|
| 597 |
+
|
| 598 |
+
with $\theta _ { v }$ being the angle between the vector $\nabla f ( x )$ and the unit vector $v$ . Equation (10) with $\cos ( \theta _ { v } ) = 1$ has a unique solution for $v$ with $\begin{array} { r } { v \ = \ \frac { \nabla f ( x ) } { | | \nabla f ( x ) | | _ { 2 } } } \end{array}$ . It follows that, if for two different directions $v , v ^ { \prime }$ we have $D _ { v } ( f ( x ) ) = D _ { v ^ { \prime } } ( f ( x ) ) \ { = } \ 1$ , then $\cos ( \theta _ { v } ) \ = \ \cos ( \theta _ { v ^ { \prime } } ) \ < \ 1$ and $| | \nabla f ( x ) | | _ { 2 } > 1$ .
|
| 599 |
+
|
| 600 |
+
# D ADDITIONAL EXPERIMENTAL RESULTS
|
| 601 |
+
|
| 602 |
+
# D.1 LEVEL SETS OF THE CRITIC
|
| 603 |
+
|
| 604 |
+
|
| 605 |
+
Figure 8: Level sets of the critic (yellow corresponds to high, purple to low values) of WGANs during training (after 10, 50, 100, 500, and 1000 iterations) on the 8Gaussian data set. Top: GPpenalty $\lambda = 1 0$ ). Middle: GP-penalty $\lambda = 1$ ). Bottom: LP-penalty $\lambda = 1 0$ ).
|
| 606 |
+
|
| 607 |
+
|
| 608 |
+
Figure 9: Level sets of the critic (yellow corresponds to high, purple to low values) of WGANs during training (after 10, 50, 100, 500, and 1000 iterations) on the 25Gaussian data set. Top: GPpenalty $\lambda = 1 0$ ). Middle: GP-penalty $\lambda = 1$ ). Bottom: LP-penalty $\lambda = 1 0$ ).
|
| 609 |
+
|
| 610 |
+
|
| 611 |
+
Figure 10: Evolution of the WGAN-GP critics loss without the regularization term $\lambda = 1$ ). Left: Median results over the 20 runs (blue area indicates quantiles, green dots outliers). Right: Single runs.
|
| 612 |
+
|
| 613 |
+
D.3 EVOLUTION OF THE EM DISTANCE
|
| 614 |
+
|
| 615 |
+
|
| 616 |
+
Figure 11: Evolution of the approximated EM distance during training WGAN-GPs with $\lambda = 1$ . Left: Median results over the 10 runs. Right: Single runs.
|
| 617 |
+
|
| 618 |
+
# D.4 DIFFERENT SAMPLING METHODS
|
| 619 |
+
|
| 620 |
+
We analyzed the effect of the GP- and the LP-penalty using different sampling procedures. In particular, we compared the sampling procedure proposed by Gulrajani et al. (2017) with variants, which generate the samples used for the regularization term by adding random noise either onto training points or onto both training and generated samples. We refer to this as “local perturbation” in the following.7 The evolution of the critics loss when using this local perturbation can be seen in Figure 12. Results are qualitatively similar to those when using the sampling procedure proposed by Gulrajani et al. (2017). Interestingly, WGAN-GP training is stabilized at a later stage if one only adds noise to training examples and not to generated examples. This indicates that enforcing the GP-penalty close to the data manifold is less harmful. However, the critic’s loss is still much more fluctuating than when training a WGAN-LP.
|
| 621 |
+
|
| 622 |
+
The evolution of the approximated EM distance when using local perturbation (by adding noise to the training examples only) is shown in Figure 13. Training with the GP-penalty leads to larger fluctuations of the approximated Wasserstein-1 distance than training with the LP-penalty. However, fluctuations are less severe compared to the setting when the GP-penalty is used in combination with the sampling procedure proposed by Gulrajani et al. (2017).
|
| 623 |
+
|
| 624 |
+
|
| 625 |
+
Figure 12: Evolution of the WGAN critic’s negative loss with local sampling (without the regularization term). Left: Median results over the 20 runs. Right: Single runs. Top: GP-penalty when generating samples by perturbing training samples only. Middle: For GP-penalty, perturbing training and generated samples. Bottom: LP-penalty, perturbing training and generated samples (very similar to perturbing only training samples)
|
| 626 |
+
|
| 627 |
+
|
| 628 |
+
Figure 13: Evolution of the approximated EM distance during training of WGANs with local perturbation $\lambda = 5$ ). Left: Median results over the 10 runs. Right: Single runs. Top: For the GP-penalty. Bottom: For the LP-penalty
|
| 629 |
+
|
| 630 |
+
# D.5 OPTIMIZING THE WASSERSTEIN-2 DISTANCE
|
| 631 |
+
|
| 632 |
+
We trained a WGAN with the objective of minimizing the Wasserstein-2 distance8, that is, with the regularization term given by
|
| 633 |
+
|
| 634 |
+
$$
|
| 635 |
+
\operatorname* { m a x } \left( \left\{ 0 , { \frac { | f ( x ) - f ( y ) | } { | | x - y | | _ { 2 } ^ { 2 } } } - 1 \right\} \right) ^ { 2 } \ ,
|
| 636 |
+
$$
|
| 637 |
+
|
| 638 |
+
and penalty weight $\lambda = 1 0$ . Results for the evolution of the critics loss and the approximated EM distance during training on the Swiss Roll data set are shown in Figure 14. Both critic loss and EM reduce smoothly, which makes the Wasserstein-2 distance (in combination with its theoretical properties) an interesting candidate to further investigations.
|
| 639 |
+
|
| 640 |
+
|
| 641 |
+
Figure 14: Evolution of the WGAN critics loss (Left) and the approximated EM distance (Right) for a WGAN-LP trained to minimize the Wasserstein-2 distance $\lambda = 1 0$ ). Shown are the medians over 5 runs.
|
| 642 |
+
|
| 643 |
+
# D.6 EXPERIMENTAL RESULTS ON CIFAR
|
| 644 |
+
|
| 645 |
+
Inception score. The inception score was proposed by Salimans et al. (2016) to evaluate the quality of images $x$ sampled from a generative model $\nu$ based on the Inception model. Let $p ( y | x )$ be the conditional probability of label $y$ for image $x$ under the Inception model and $\begin{array} { r } { p ( y ) = \int p ( y | x ) \nu ( x ) d x } \end{array}$ the marginal probability of labels $y$ with respect to samples generated from $\nu$ . Then the Inception score is given by
|
| 646 |
+
|
| 647 |
+
$$
|
| 648 |
+
\exp \left( \mathbb { E } _ { x \sim \nu } [ K L ( p ( y | x ) , p ( y ) ] ) \right) \ .
|
| 649 |
+
$$
|
| 650 |
+
|
| 651 |
+
Intuitively, a good generative model should produce samples for which the conditional label distribution has low entropy, while the variability over samples and thus the entropy of the marginal label distribution should be high. Therefore, a higher Inception score indicates a better performance of the generative model.
|
| 652 |
+
|
| 653 |
+
The maximal Inception scores reported in Table 1 are representative for the general evolution of the scores for WGAN-LP and WGAN-GP during training. As an example we show the evolution of the Inception score for penalty weights of $\lambda = 5$ and $\lambda = 1 0 0$ in Figure 15. It becomes clear that WGAN-GP performs similar to WGAN-LP for small values of the regularization parameter but much worse for larger values (this was consistently observed in all experiments). In Figure 16 we compare the performance of WGAN-LP and WGAN-GP in terms of the critics loss on a separate validation set, which again demonstrates a more stable behavior for WGAN-LP with respect to the choice of lambda.
|
| 654 |
+
|
| 655 |
+
|
| 656 |
+
Figure 15: Evolution of Inception score on CIFAR for WGAN-LP in blue (solid) and WGAN-GP in red (dotted). Left: for regularization parameter $\lambda = 5$ . Right: for regularization parameter $\lambda = 1 0 0$ .
|
| 657 |
+
|
| 658 |
+
We also trained WGAN-GP and WGAN-LP with a conditional model (making use of the label information of CIFAR10) with $\lambda = 1 0$ and found a similar performance for both, i.e. $8 . 5 3 7 \pm 0 . 1 3 3$ and $8 . 4 6 2 \pm 0 . 1 1 5$ for WGAN-GP and WGAN-LP, respectively.
|
| 659 |
+
|
| 660 |
+
|
| 661 |
+
Figure 16: Evolution of validation loss on CIFAR. Black/purple curves indicate the total loss, blue curves the loss without regularization term, and red the regularization term only. Light colored curves indicate the true values, dark solid lines the average over a window of 5 iterations. Left: WGAN-GP. Right: WGAN-LP. Top: with $\lambda = 5$ . Bottom: with $\lambda = 1 0 0$ .
|
| 662 |
+
|
| 663 |
+
# D.7 RELATED PENALTIES
|
| 664 |
+
|
| 665 |
+
Level sets for WGANs trained with the regularization terms given by Equation (7) and (9) and penalty weight 10 are shown in Figure 17. As the evolution of the level sets and the sampled points indicate, training properly converges. However, on CIFAR-10, the same penalties did not lead to good results. As shown in Figure 18, using (7) for regularization initially lead to improving Inception scores but then quickly started to diverge, while using (9) lead to even greater instability.
|
| 666 |
+
|
| 667 |
+
|
| 668 |
+
Figure 17: Level sets of the critic $f$ of WGANs during training, after 10, 50, 100, 500, and 1000 iterations. Yellow corresponds to high, purple to low values of $f$ . Training samples are indicated in red, generated samples in blue. Top: With the regularization term given in Equation (7) and $\lambda = 1 0$ . Bottom: With the regularization term given in Equation (9) and $\lambda = 1 0$ .
|
| 669 |
+
|
| 670 |
+
|
| 671 |
+
Figure 18: Inception scores for regularization Equation (7) for penalty weights 100 (red) and 5 (blue), shown on the left, and Inception scores for training with the regularization Equation (9) for penalty weights 100 (red) and 5 (blue), shown on the right.
|
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parse/train/B1l08oAct7/B1l08oAct7_layout.pdf
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| 1 |
+
# HUMAN PERCEPTION IN COMPUTER VISION / CONFERENCE SUBMISSIONS
|
| 2 |
+
|
| 3 |
+
# Ron Dekel ∗
|
| 4 |
+
|
| 5 |
+
Department of Neurobiology Weizmann Institute of Science Rehovot, PA 7610001, Israel ron.dekel@weizmann.ac.il
|
| 6 |
+
|
| 7 |
+
# ABSTRACT
|
| 8 |
+
|
| 9 |
+
Computer vision has made remarkable progress in recent years. Deep neural network (DNN) models optimized to identify objects in images exhibit unprecedented task-trained accuracy and, remarkably, some generalization ability: new visual problems can now be solved more easily based on previous learning. Biological vision (learned in life and through evolution) is also accurate and generalpurpose. Is it possible that these different learning regimes converge to similar problem-dependent optimal computations? We therefore asked whether the human system-level computation of visual perception has DNN correlates and considered several anecdotal test cases. We found that perceptual sensitivity to image changes has DNN mid-computation correlates, while sensitivity to segmentation, crowding and shape has DNN end-computation correlates. Our results quantify the applicability of using DNN computation to estimate perceptual loss, and are consistent with the fascinating theoretical view that properties of human perception are a consequence of architecture-independent visual learning.
|
| 10 |
+
|
| 11 |
+
# 1 QUICK EXPERT SUMMARY
|
| 12 |
+
|
| 13 |
+
Considering the learned computation of ImageNet-trained DNNs, we find:
|
| 14 |
+
|
| 15 |
+
• Large computation changes for perceptually salient image changes (Figure 1).
|
| 16 |
+
• Gestalt: segmentation, crowding, and shape interactions in computation (Figure 2).
|
| 17 |
+
• Contrast constancy: bandpass transduction in first layers is later corrected (Figure 3).
|
| 18 |
+
|
| 19 |
+
These properties are reminiscent of human perception, perhaps because learned general-purpose classifiers (human and DNN) tend to converge.
|
| 20 |
+
|
| 21 |
+
# 2 INTRODUCTION
|
| 22 |
+
|
| 23 |
+
Deep neural networks (DNNs) are a class of computer learning algorithms that have become widely used in recent years (LeCun et al., 2015). By training with millions of examples, such models achieve unparalleled degrees of task-trained accuracy (Krizhevsky et al., 2012). This is not unprecedented on its own - steady progress has been made in computer vision for decades, and to some degree current designs are just scaled versions of long-known principles (Lecun et al., 1998). In previous models, however, only the design is general-purpose, while learning is mostly specific to the context of a trained task. Interestingly, for current DNNs trained to solve a large-scale image recognition problem (Russakovsky et al., 2014), the learned computation is useful as a building block for drastically different and untrained visual problems (Huh et al., 2016; Yosinski et al., 2014).
|
| 24 |
+
|
| 25 |
+
For example, orientation- and frequency-selective features (Gabor patches) can be considered general-purpose visual computations. Such features are routinely discovered by DNNs (Krizhevsky et al., 2012; Zeiler & Fergus, 2013), by other learning algorithms (Hinton & Salakhutdinov, 2006;
|
| 26 |
+
|
| 27 |
+
Lee et al., 2008; 2009; Olshausen & Field, 1997), and are extensively hard-coded in computer vision (Jain & Farrokhnia, 1991). Furthermore, a similar computation is believed to underlie the spatial response properties of visual neurons of diverse animal phyla (Carandini et al., 2005; DeAngelis et al., 1995; Hubel & Wiesel, 1968; Seelig & Jayaraman, 2013), and is evident in human visual perception (Campbell & Robson, 1968; Fogel & Sagi, 1989; Neri et al., 1999). This diversity culminates in satisfying theoretical arguments as to why Gabor-like features are so useful in general-purpose vision (Olshausen, 1996; Olshausen & Field, 1997).
|
| 28 |
+
|
| 29 |
+
As an extension, general-purpose computations are perhaps of universal use. For example, a dimensionality reduction transformation that optimally preserves recognition-relevant information may constitute an ideal computation for both DNN and animal. More formally, different learning algorithms with different physical implementations may converge to the same computation when similar (or sufficiently general) problems are solved near-optimally. Following this line of reasoning, DNN models with good general-purpose computations may be computationally similar to biological visual systems, even more so than less accurate and less general biologically plausible simulations (Kriegeskorte, 2015; Yamins & DiCarlo, 2016).
|
| 30 |
+
|
| 31 |
+
Related work seems to be consistent with computation convergence. First, different DNN training regimes seem to converge to a similar learned computation (Li et al., 2015; Zhou et al., 2014). Second, image representation may be similar in trained DNN and in biological visual systems. That is, when the same images are processed by DNN and by humans or monkeys, the final DNN computation stages are strong predictors of human fMRI and monkey electrophysiology data collected from visual areas V4 and IT (Cadieu et al., 2014; Khaligh-Razavi & Kriegeskorte, 2014; Yamins et al., 2014). Furthermore, more accurate DNN models exhibit stronger predictive power (Cadieu et al., 2014; Dubey & Agarwal, 2016; Yamins et al., 2014), and the final DNN computation stage is even a strong predictor of human-perceived shape discrimination (Kubilius et al., 2016). However, some caution is perhaps unavoidable, since measured similarity may be confounded with categorization consistency, view-invariance resilience, or similarity in the inherent difficulty of the tasks undergoing comparison. A complementary approach is to consider images that were produced by optimizing trained DNN-based perceptual metrics (Gatys et al., 2015a;b; Johnson et al., 2016; Ledig et al., 2016), which perhaps yields undeniable evidence of non-trivial computational similarity, although a more objective approach may be warranted.
|
| 32 |
+
|
| 33 |
+
Here, we quantify the similarity between human visual perception, as measured by psychophysical experiments, and individual computational stages (layers) in feed-forward DNNs trained on a large-scale image recognition problem (ImageNet LSVRC). Comparison is achieved by feeding the experimental image stimuli to the trained DNN and comparing a DNN metric (mean mutual information or mean absolute change) to perceptual data. The use of reduced (simplified and typically non-natural) stimuli ensures identical inherent task difficulty across compared categories and prevents confounding of categorization consistency with measured similarity. Perception, a systemlevel computation, may be influenced less by the architectural discrepancy (biology vs. DNN) than are neural recordings.
|
| 34 |
+
|
| 35 |
+
# 3 CORRELATE FOR IMAGE CHANGE SENSITIVITY
|
| 36 |
+
|
| 37 |
+
From a perceptual perspective, an image change of fixed size has different saliency depending on image context (Polat & Sagi, 1993). To investigate whether the computation in trained DNNs exhibits similar contextual modulation, we used the Local Image Masking Database (Alam et al., 2014), in which 1080 partially-overlapping images were subjected to different levels of the same random additive noise perturbation, and for each image, a psychophysical experiment determined the threshold noise level at which the added-noise image is discriminated from two noiseless copies at $7 5 \%$ (Figure 1a). Threshold is the objective function that is compared with an $L _ { 1 }$ -distance correlate in the DNN representation. The scale of measured threshold was:
|
| 38 |
+
|
| 39 |
+
$$
|
| 40 |
+
2 0 \cdot \log _ { 1 0 } \left( \frac { \mathrm { s t d } \left( n o i s e \right) } { T } \right) ,
|
| 41 |
+
$$
|
| 42 |
+
|
| 43 |
+
where std $( n o i s e )$ is the standard deviation of the additive noise, and $T$ is the mean image pixel value calculated over the region where the noise is added (i.e. image center).
|
| 44 |
+
|
| 45 |
+
|
| 46 |
+
Figure 1: Predicting perturbation thresholds. a, For a fixed image perturbation, perceptual detection threshold (visualized by red arrow) depends on image context. b, Measured perceptual threshold is correlated with the average $L _ { 1 }$ change in DNN computation due to image perturbation (for DNN model VGG-19, image scale $= 1 0 0 \%$ ). c, Explained variability $( R ^ { 2 } )$ of perceptual threshold data when $L _ { 1 }$ change is based on isolated computational layers for different input image scales. Same VGG-19 model as in (b). X-axis labels: data refers to raw image pixel data, $\mathsf { c o n v ^ { * } } _ { - 1 }$ and ${ \mathsf { f c } } _ { - } *$ are the before-ReLU output of a convolution and a fully-connected operation, respectively, and prob is the output class label probabilities vector. d, Example images for whcih predicted threshold in b is much higher than perceptually measured (”Overshoot”, where perturbation saliency is better than predicted), or vise versa (”Undershoot”). Examples are considered from several perceptual threshold ranges $\pm 2 \mathrm { d B }$ of shown number).
|
| 47 |
+
|
| 48 |
+
The DNN correlate of perceptual threshold we used was the average $L _ { 1 }$ change in DNN computation between added-noise images and the original, noiseless image. Formally,
|
| 49 |
+
|
| 50 |
+
$$
|
| 51 |
+
L _ { 1 } ^ { i , n } ( I ) = \left| \overline { { { a _ { i } \left( I + n o i s e \left( n \right) \right) } } } - a _ { i } \left( I \right) \right| ,
|
| 52 |
+
$$
|
| 53 |
+
|
| 54 |
+
where $a _ { i } \left( X \right)$ is the activation value of neuron $i$ during the DNN feedforward pass for input image $X$ , and the inner average (denoted by bar) is taken over repetitions with random $n$ -sized noise (noise is introduced at random phase spectra in a fixed image location, an augmentation that follows the between-image randomization described by Alam et al., 2014; the number of repetitions was 10 or more). Unless otherwise specified, the final $L _ { 1 }$ prediction is $L _ { 1 } ^ { i , n }$ averaged across noise levels $( - 4 0$ to $2 5 { \mathrm { ~ d B } }$ with 5-dB intervals) and computational neurons (first within and then across computational stages). Using $L _ { 1 }$ averaged across noise levels as a correlate for the noise level of perceptual threshold is a simple approximation with minimal assumptions.
|
| 55 |
+
|
| 56 |
+
Results show that the $L _ { 1 }$ metric is correlated with the perceptual threshold for all tested DNN architectures (Figure 1b, 4a-c). In other words, higher values of the $L _ { 1 }$ metric (indicating larger changes in DNN computation due to image perturbation, consistent with higher perturbation saliency) are correlated with lower values of measured perceptual threshold (indicating that weaker noise levels are detectable, i.e. higher saliency once more).
|
| 57 |
+
|
| 58 |
+
Table 1: Prediction accuracy. Percent of linearly explained variability $( R ^ { 2 } )$ , absolute value of Spearman rank-order correlation coefficient (SROCC), and the root mean squared error of the linear prediction (RMSE) are presented for each prediction model. Note the measurement scale of the threshold data being predicted (Eq. 1). $( ^ { * } )$ Thresholds linearized through a logistic transform before prediction (see Larson & Chandler, 2010), possibly increasing but not decreasing measured predictive strength. $( ^ { * * } )$ Average of four similar alternatives.
|
| 59 |
+
|
| 60 |
+
<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td></tr><tr><td>Signal-noise ratio</td><td></td><td></td><td></td></tr><tr><td>Spectral change</td><td>.20 .25</td><td>.39</td><td>7.67</td></tr><tr><td>RMS contrast (Alam et al., 2014)</td><td>.27*</td><td>.61</td><td>7.42</td></tr><tr><td>L1 VGG-19 (50%)</td><td>.57</td><td>.46</td><td>1</td></tr><tr><td>L1 VGG-19 (66%)</td><td>.60</td><td>.77</td><td>5.57 5.42</td></tr><tr><td>L1 VGG-19 (100%)</td><td>.60</td><td>.79</td><td></td></tr><tr><td>Perceptual model** (Alam et al., 2014)</td><td>.60*</td><td>.79</td><td>5.40</td></tr><tr><td></td><td></td><td>.70</td><td>5.73</td></tr><tr><td>Inter-person (Alam et al., 2014)</td><td>.84*</td><td>.87</td><td>4.08</td></tr></table>
|
| 61 |
+
|
| 62 |
+
To quantify and compare predictive power, we considered the percent of linearly explained variability $( R ^ { 2 } )$ . For all tested DNN architectures, the prediction explains about $6 0 \%$ of the perceptual variability (Tables 1, 2; baselines at Tables 3-5), where inter-person similarity representing theoretical maximum is $84 \%$ (Alam et al., 2014). The DNN prediction is far more accurate than a prediction based on simple image statistical properties (e.g. RMS contrast), and is on par with a detailed perceptual model that relies on dozens of psychophysically collected parameters (Alam et al., 2014). The Spearmann correlation coefficient is much higher compared with the perceptual model (with an absolute SROCC value of about 0.79 compared with 0.70, Table 1), suggesting that the $L _ { 1 }$ metric gets the order right but not the scale. We did not compare these results with models that fit the experimental data (e.g. Alam et al., 2015; Liu & Allebach, 2016), since the $L _ { 1 }$ metric has no explicit parameters. Also, different DNN architectures exhibited high similarity in their predictions $\bar { R } ^ { 2 }$ of about 0.9, e.g. Figure 4d).
|
| 63 |
+
|
| 64 |
+
Prediction can also be made from isolated computational stages, instead of across all stages as before. This analysis shows that the predictive power peaks mid-computation across all tested image scales (Figure 1c). This peak is consistent with use of middle DNN layers to optimize perceptual metrics (Gatys et al., 2015a;b; Ledig et al., 2016), and is reminiscent of cases in which low- to mid-level vision is the performance limiting computation in the detection of at-threshold stimuli (Campbell & Robson, 1968; Del Cul et al., 2007).
|
| 65 |
+
|
| 66 |
+
Finally, considering the images for which the $L _ { 1 }$ -based prediction has a high error suggests a factor which causes a systematic inconsistency with perception (Figures 1d, 6). This factor may be related to the mean image luminance: by introducing noise perturbations according to the scale of Equation 1, a fixed noise size (in dB) corresponds to smaller pixel changes in dark compared with bright images. (Using this scales reflects an assumption of multiplicative rather than additive conservation; this assumption may be justified for the representation at the final but perhaps not the intermediate computational stages considering the log-linear contrast response discussed in Section 5). Another factor may the degree to which image content is identifiable.
|
| 67 |
+
|
| 68 |
+
The previous analysis suggested gross computational similarity between human perception and trained DNNs. Next, we aimed to extend the comparison to more interpretable properties of perception by considering more highly controlled designs. To this end, we considered cases in which a static background context modulates the difficulty of discriminating a foreground shape, despite no spatial overlap of foreground and background. This permits interpretation by considering the cause of the modulation.
|
| 69 |
+
|
| 70 |
+
We first consider segmentation, in which arrangement is better discriminated for arrays of consistently oriented lines compared with inconsistently oriented lines (Figure 2a) (Pinchuk-Yacobi et al., 2016). Crowding is considered next, where surround clutter that is similar to the discriminated target leads to deteriorated discrimination performance (Figure 2b) (Livne & Sagi, 2007). Last to be addressed is object superiority, in which a target line location is better discriminated when it is in a shape-forming layout (Figure 2c) (Weisstein & Harris, 1974). In this case, clutter is controlled by having the same fixed number of lines in context. To measure perceptual discrimination, these works introduced performance-limiting manipulations such as location jittering, brief presentation, and temporal masking. While different manipulations showed different measured values, order-of-difficulty was typically preserved. Here we changed all the original performance-limiting manipulations to location jittering (whole-shape or element-wise, see Section 8.4).
|
| 71 |
+
|
| 72 |
+
To quantify discrimination difficulty in DNNs, we measured the target-discriminative information of isolated neurons (where performance is limited by location jittering noise), then averaged across all neurons (first within and then across computational layer stages). Specifically, for each neuron, we measured the reduction in categorization uncertainty due to observation, termed mutual information (MI):
|
| 73 |
+
|
| 74 |
+
$$
|
| 75 |
+
M I \left( A _ { i } ; C \right) = H \left( C \right) - H \left( C \vert A _ { i } \right) ,
|
| 76 |
+
$$
|
| 77 |
+
|
| 78 |
+
where H stands for entropy, and $A _ { i }$ is a random variable for the value of neuron i when the DNN processes a random image from a category defined by the random variable C. For example, if a neuron gives a value in the range of 100.0 to 200.0 when the DNN processes images from category A, and 300.0 to 400.0 for category B, then the category is always known by observing the value, and so mutual information is high $\mathbf { M } \mathbf { I } { = } 1$ bits). On the other extreme, if the neuron has no discriminative task information, then ${ \bf M I } { = } 0$ bits. To measure MI, we quantized activations into eight equal-amount bins, and used 500 samples (repetitions having different location jittering noise) across categories. The motivation for this correlate is the assumption that the perceptual order-of-difficulty reflects the quantity of task-discriminative information in the representation.
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Results show that, across hundreds of configurations (varying pattern element size, target location, jitter magnitude, and DNN architecture; see Section 8.4), the qualitative order of difficulty in terms of the DNN MI metric is consistent with the order of difficulty measured in human psychophysical experiments, for the conditions addressing segmentation and crowding (Figures 2d, 7; for baseline models see Figure 8). It is interesting to note that the increase in similarity develops gradually along different layer types in the DNN computation (i.e. not just pooling layers), and is accompanied by a gradual increase in the quantity of task-relevant information (Figure 2e-g). This indicates a link between task relevance and computational similarity for the tested conditions. Note that unlike the evident increase in isolated unit task information, the task information from all units combined decreases by definition along any computational hierarchy. An intuition for this result is that the total hidden information decreases, while more accessible per-unit information increases.
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For shape formation, four out of six shapes consistently show order of difficulty like perception, and two shapes consistently do no (caricature at Figure 2h; actual data at Figure 9).
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Figure 2: Background context. a-c, Illustrations of reproduced discrimination stimuli for three psychophysical experiments (actual images used were white-on-black rather than black-on-white, and pattern size was smaller, see Figures 12-14). d, Number of configurations for which orderof-difficulty in discrimination is qualitatively consistency with perception according to a mutual information DNN metric. Configurations vary in pattern (element size, target location, and jitter magnitude; see Section 8.4) and in DNN architecture used (CaffeNet, GoogLeNet, VGG-19, and ResNet-152). DNN metric is the average across neurons of the isolated neuron target-discriminative information (averaged first within, and then across computational layer stages), where performance is limited by location jittering (e.g. evident jitter in illustrations). e-g, The value of the MI metric across computational layers of model VGG-19 for a typical pattern configuration. The six ”hard” (gray) lines in Shape MI correspond to six different layouts (see Section 8.4.3). Analysis shows that for isolated computation stages, similarity to perception is evident only at the final DNN computation stages. h, A caricature summarizing the similarity and discrepancy of perception and the MI-based DNN prediction for Shape (see Figure 9).
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# 5 CORRELATE FOR CONTRAST SENSITIVITY
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A cornerstone of biological vision research is the use of sine gratings at different frequencies, orientations, and contrasts (Campbell & Robson, 1968). Notable are results showing that the lowest perceivable contrast in human perception depends on frequency. Specifically, high spatial frequencies are attenuated by the optics of the eye, and low spatial frequencies are believed to be attenuated due to processing inefficiencies (Watson & Ahumada, 2008), so that the lowest perceivable contrast is found at intermediate frequencies. (To appreciate this yourself, examine Figure 3a). Thus, for low-contrast gratings, the physical quantity of contrast is not perceived correctly: it is not preserved across spatial frequencies. Interestingly, this is corrected for gratings of higher contrasts, for which perceived contrast is more constant across spatial frequencies (Georgeson & Sullivan, 1975).
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The DNN correlate we considered is the mean absolute change in DNN representation between a gray image and sinusoidal gratings, at all combinations of spatial frequency and contrast. Formally, for neurons in a given layer, we measured:
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$$
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L _ { 1 } ( c o n t r a s t , f r e q u e n c y ) = \frac { 1 } { N _ { n e u r o n s } } \sum _ { i = 1 } ^ { N _ { n e u r o n s } } \left| \overline { { a _ { i } \left( c o n t r a s t , f r e q u e n c y \right) } } - a _ { i } \left( 0 , 0 \right) \right| ,
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$$
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where $a _ { i }$ (contrast, frequency) is the average activation value of neuron $i$ to 250 sine images (random orientation, random phase), $a _ { i } \left( 0 , 0 \right)$ is the response to a blank (gray) image, and $N _ { n }$ eurons is the number of neurons in the layer. This measure reflects the overall change in response vs. the gray image.
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Results show a bandpass response for low-contrast gratings (blue lines strongly modulated by frequency, Figures 3, 10), and what appears to be a mostly constant response at high contrast for end-computation layers (red lines appear more invariant to frequency), in accordance with perception.
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We next aimed to compare these results with perception. Data from human experiments is generally iso-output (i.e. for a pre-set output, such as $7 5 \%$ detection accuracy, the input is varied to find the value which produce the preset output). However, the DNN measurements here are iso-input (i.e. for a fixed input contrast the $L _ { 1 }$ is measured). As such, human data should be compared to the interpoalted inverse of DNN measurements. Specifically, for a set output value, the interpolated contrast value which produce the output is found for every frequency (Figure 11). This analysis permits quantifying the similarity of iso-output curves for human and DNN, measured here as the percent of log-Contrast variability in human measurements which is explained by the DNN predictions. This showed a high explained variability at the end computation stage (prob layer, $\mathbf { \dot { \mathit { R } } ^ { 2 } = 9 4 \% }$ , but importantly, a similarly high value at the first computational stage (conv1 1 layer, $R ^ { 2 } = 9 6 \%$ ). Intiutively, while the ”internal representation” variability in terms of $L _ { 1 }$ is small, the iso-output number-of-input-contrast-cahnges variability is still high. For example. for the prob layer, about the same $L _ { 1 }$ is measured for (Contrast $^ { - 1 }$ ,freq $= 7 5$ ) and for (Contras ${ = } 0 . 1 8$ ,freq $= 1 2$ ).
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An interesting, unexpected observation is that the logarithmically spaced contrast inputs are linearly spaced at the end-computation layers. That is, the average change in DNN representation scales logarithmically with the size of input change. This can be quantified by the correlation of output $L _ { 1 }$ with log Contrast input, which showed $R ^ { 2 } = 9 8 \%$ (averaged across spatial frequencies) for prob, while much lower values were observed for early and middle layers (up to layer fc7). The same computation when scrambling the learned parameters of the model showed $R ^ { 2 } = 6 0 \%$ . Because the degree of log-linearity observed was extremely high, it may be an important emergent property of the learned DNN computation, which may deserve further investigation. However, this property is only reminiscent and not immediately consistent with the perceptual power-law scaling (Gottesman et al., 1981).
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Figure 3: Contrast sensitivity. a. Perceived contrast is strongly affected by spatial frequency at low contrast, but less so at high contrast (which preserves the physical quantity of contrast and thus termed constancy). b. The $L _ { 1 }$ change in VGG-19 representation between a gray image and images depicting sinusoidal gratings at each combination of sine spatial frequency $\mathbf { \dot { x } }$ -axis) and contrast (color) (random orientation, random phase), considering the raw image pixel data representation (data), the before-ReLU output of the first convolutional layer representation (conv1 1), the output of the last fully-connected layer representation (fc8), and the output class label probabilities representation (prob).
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# 6 DISCUSSION
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# 6.1 HUMAN PERCEPTION IN COMPUTER VISION
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It may be tempting to believe that what we see is the result of a simple transformation of visual input. Centuries of psychophysics have, however, revealed complex properties in perception, by crafting stimuli that isolate different perceptual properties. In our study, we used the same stimuli to investigate the learned properties of deep neural networks (DNNs), which are the leading computer vision algorithms to date (LeCun et al., 2015).
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The DNNs we used were trained in a supervised fashion to assign labels to input images. To some degree, this task resembles the simple verbal explanations given to children by their parents. Since human perception is obviously much richer than the simple external supervision provided, we were not surprised to find that the best correlate for perceptual saliency of image changes is a part of the DNN computation that is only supervised indirectly (i.e. the mid-computation stage). This similarity is so strong, that even with no fine-tuning to human perception, the DNN metric is competitively accurate, even compared with a direct model of perception.
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This strong, quantifiable similarity to a gross aspect of perception may, however, reflect a mix of similarities and discrepancies in different perceptual properties. To address isolated perceptual effects, we considered experiments that manipulate a spatial interaction, where the difficulty of discriminating a foreground target is modulated by a background context. Results showed modulation of DNN target diagnostic, isolated unit information, consistent with the modulation found in perceptual discrimination. This was shown for contextual interactions reflecting grouping/segmentation (Harris et al., 2015), crowding/clutter (Livne & Sagi, 2007; Pelli et al., 2004), and shape superiority (Weisstein & Harris, 1974). DNN similarity to these groupings/gestalt phenomena appeared at the end-computation stages.
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No less interesting, are the cases in which there is no similarity. For example, perceptual effects related to 3D (Erdogan & Jacobs, 2016) and symmetry (Pramod & Arun, 2016) do not appear to have a strong correlate in the DNN computation. Indeed, it may be interesting to investigate the influence of visual experience in these cases. And, equally important, similarity should be considered in terms of specific perceptual properties rather than as a general statement.
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# 6.2 RECURRENT VS. FEEDFORWARD CONNECTIVITY
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In the human hierarchy of visual processing areas, information is believed to be processed in a feedforward sweep, followed by recurrent processing loops (top-down and lateral) (Lamme & Roelfsema, 2000). Thus, for example, the early visual areas can perform deep computations. Since mapping from visual areas to DNN computational layers is not simple, it will not be considered here. (Note that ResNet connectivity is perhaps reminiscent of unrolled recurrent processing).
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Interestingly, debate is ongoing about the degree to which visual perception is dependent on recurrent connectivity (Fabre-Thorpe et al., 1998; Hung et al., 2005): recurrent representations are obviously richer, but feedforward computations converge much faster. An implicit question here regarding the extent of feasible feed-forward representations is, perhaps: Can contour segmentation, contextual influences, and complex shapes be learned? Based on the results reported here for feedforward DNNs, a feedforward representation may seem sufficient. However, the extent to which this is true may be very limited. In this study we used small images with a small number of lines, while effects such as contour integration seem to take place even in very large configurations (Field et al., 1993). Such scaling seems more likely in a recurrent implementation. As such, a reasonable hypothesis may be that the full extent of contextual influence is only realizable with recurrence, while feedforward DNNs learn a limited version by converging towards a useful computation.
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# 6.3 IMPLICATIONS AND FUTURE WORK
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# 6.3.1 USE IN BRAIN MODELING
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The use of DNNs in modeling of visual perception (or of biological visual systems in general) is subject to a tradeoff between accuracy and biological plausibility. In terms of architecture, other deep models better approximate our current understanding of the visual system (Riesenhuber &
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Poggio, 1999; Serre, 2014). However, the computation in trained DNN models is quite generalpurpose (Huh et al., 2016; Yosinski et al., 2014) and offers unparalleled accuracy in recognition tasks (LeCun et al., 2015). Since visual computations are, to some degree, task- rather than architecturedependent, an accurate and general-purpose DNN model may better resemble biological processing than less accurate biologically plausible ones (Kriegeskorte, 2015; Yamins & DiCarlo, 2016). We support this view by considering a controlled condition in which similarity is not confounded with task difficulty or categorization consistency.
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# 6.3.2 USE IN PSYCHOPHYSICS
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Our results imply that trained DNN models have good predictive value for outcomes of psychophysical experiments, permitting a zero-cost first-order approximation. Note, however, that the scope of such simulations may be limited, since learning (Sagi, 2011) and adaptation (Webster, 2011) were not considered here.
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Another fascinating option is the formation of hypotheses in terms of mathematically differentiable trained-DNN constraints, whereby it is possible to efficiently solve for the visual stimuli that optimally dissociate the hypotheses (see Gatys et al. 2015a;b; Mordvintsev et al. 2015 and note Goodfellow et al. 2014; Szegedy et al. 2013). The conclusions drawn from such stimuli can be independent of the theoretical assumptions about the generating process (for example, creating new visual illusions that can be seen regardless of how they were created).
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# 6.3.3 USE IN ENGINEERING (A PERCEPTUAL LOSS METRIC)
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As proposed previously (Dosovitskiy & Brox, 2016; Johnson et al., 2016; Ledig et al., 2016), the saliency of small image changes can be estimated as the representational distance in trained DNNs. Here, we quantified this approach by relying on data from a controlled psychophysical experiment (Alam et al., 2014). We found the metric to be far superior to simple image statistical properties, and on par with a detailed perceptual model (Alam et al., 2014). This metric can be useful in image compression, whereby optimizing degradation across image sub-patches by comparing perceptual loss may minimize visual artifacts and content loss.
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# ACKNOWLEDGMENTS
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We thank Yoram Bonneh for his valuable questions which led to much of this work.
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Dov Sagi. Perceptual learning in vision research. Vision research, 51(13):1552–1566, 2011. ISSN 0042-6989.
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Karen Simonyan and Andrew Zisserman. Very Deep Convolutional Networks for Large-Scale Image Recognition. sep 2014.
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Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.
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Andrea Vedaldi and Karel Lenc. Matconvnet: Convolutional neural networks for matlab. In Proceedings of the 23rd ACM international conference on Multimedia, pp. 689–692. ACM, 2015. ISBN 1450334598.
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Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pp. 3320–3328, 2014.
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Matthew D Zeiler and Rob Fergus. Visualizing and Understanding Convolutional Networks. nov 2013.
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Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Object Detectors Emerge in Deep Scene CNNs. pp. 12, dec 2014.
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Figure 4: Predicting perceptual sensitivity to image changes (following Figure 1). a-c, The $L _ { 1 }$ change in CaffeNet, GoogLeNet, and ResNet-152 DNN architectures as a function of perceptual threshold. d, The $L _ { 1 }$ change in GoogLeNet as a function of the $L _ { 1 }$ change in VGG-19.
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Figure 5: Prediction accuracy as a function of computational stage. a, Predicting perceptual sensitivity for model VGG-19 using the best single kernel (i.e. using one fitting parameter, no cross validation), vs. the standard $L _ { 1 }$ metric (reproduced from Figure 1). b, For non-branch computational stages of model ResNet-152.
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Table 2: Accuracy of perceptual sensitivity prediction and task-trained ImageNet center-crop top-1 validation accuracy for different DNN models (following Table 1 from which third row is reproduced; used scale: $100 \%$ ). The quality of prediction for ResNet-152 improves dramatically if only the first tens of layers are considered (see Figure 5b).
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<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td><td>Recognition accuracy</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CaffeNet</td><td>.59</td><td>.78</td><td>5.44</td><td>56%</td></tr><tr><td>GoogLeNet</td><td>.59</td><td>.79</td><td>5.45 5.40</td><td>66%</td></tr><tr><td>VGG-19</td><td>.60</td><td>.79</td><td>5.82</td><td>70%</td></tr><tr><td>ResNet-152</td><td>.53</td><td>.74</td><td></td><td>75%</td></tr></table>
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Table 3: Accuracy of perceptual sensitivity prediction for baseline models (see Section 8.2; used scale: $100 \%$ ).
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| 299 |
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<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>VGG-19, scrambled weights</td><td>.18</td><td>.39</td><td>7.76</td></tr><tr><td>Gabor filter bank</td><td>.32</td><td>.12</td><td>8.03</td></tr><tr><td>Steerable-pyramid filter bank</td><td>.37</td><td>.15</td><td>7.91</td></tr></table>
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Table 4: Accuracy of perceptual sensitivity prediction during CaffeNet model standard training (used scale: $100 \%$ ). Last row reproduced from Table 2.
|
| 303 |
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|
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<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td><td>Recognition accuracy</td></tr><tr><td>CaffeNetiter1</td><td></td><td></td><td>6.30</td><td></td></tr><tr><td>CaffeNetiter50K</td><td>.46 .59</td><td>.67 .79</td><td>5.43</td><td>0% 37%</td></tr><tr><td>CaffeNetiter100K</td><td>.60</td><td>.79</td><td>5.41</td><td>39%</td></tr><tr><td>CaffeNetiter150K</td><td>.60</td><td>.78</td><td>5.43</td><td>53%</td></tr><tr><td>CaffeNetiter200K</td><td>.59</td><td>.78</td><td>5.45</td><td>54%</td></tr><tr><td>CaffeNetiter250K</td><td>.59</td><td>.78</td><td>5.43</td><td>56%</td></tr><tr><td>CaffeNetiter300K</td><td>.59</td><td>.78</td><td>5.44</td><td>56%</td></tr><tr><td>CaffeNetiter310K</td><td>.59</td><td>.78</td><td>5.44</td><td>56%</td></tr></table>
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| 305 |
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Table 5: Robustness of perceptual sensitivity prediction for varying prediction parameters for model VGG-19. First three rows reproduced from Table 1. Measurements for the lower noise range of -60:-40 dB were omitted by mistake.
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| 307 |
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<table><tr><td>Scale</td><td>Metric</td><td>Augmentation</td><td>Noise range</td><td>R²</td><td>SROCC</td><td>RMSE</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>100%</td><td>L1</td><td>noise phase</td><td>-40:25 dB</td><td>.60</td><td>.79</td><td>5.40</td></tr><tr><td>66%</td><td>L1</td><td>noise phase</td><td>-40:25 dB</td><td>.60</td><td>.79</td><td>5.42</td></tr><tr><td>50%</td><td>L1</td><td>noise phase</td><td>-40:25 dB</td><td>.57</td><td>.77</td><td>5.57</td></tr><tr><td>100%</td><td>L2</td><td>noise phase</td><td>-40:25 dB</td><td>.62</td><td>.80</td><td>5.29</td></tr><tr><td>100%</td><td>L1</td><td>None</td><td>-40:25 dB</td><td>.58</td><td>.77</td><td>5.55</td></tr><tr><td>100%</td><td>L1</td><td>noise phase</td><td>-20:25 dB</td><td>.59</td><td>.78</td><td>5.46</td></tr><tr><td>100%</td><td>L1</td><td>noise phase</td><td>-40:5 dB</td><td>.59</td><td>.79</td><td>5.43</td></tr></table>
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Table 6: Background context for Shape. Shown is the Spearmann correlation coefficient (SROCC) of perceptual data vs. model-based MI prediction across shapes (i.e. considering all shapes rather than only Easy vs. Hard; note that the original robust finding the superiority of the Easy shape). Perceptual data from Weisstein & Harris (1974), where ”Day 1” and ”Days $2 { - } 4 ^ { \dag }$ (averaged) are for the reduced-masking condition depicted in their Figure 3.)
|
| 311 |
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<table><tr><td>Model</td><td></td><td>Day 1 Days 2-4</td><td>Masked</td></tr><tr><td>VGG-19</td><td>.36</td><td></td><td>.15</td></tr><tr><td>GoogLeNet</td><td>.31</td><td>.37 .22</td><td>.16</td></tr><tr><td>MRSA-152</td><td>.26</td><td>.26</td><td>.11</td></tr><tr><td>CaffeNet iter 1</td><td>.32</td><td></td><td>.39</td></tr><tr><td>CaffeNet iter 50K</td><td>.15</td><td>.29</td><td></td></tr><tr><td>CaffeNetiter310K</td><td>.16</td><td>.19</td><td>.16 .18</td></tr><tr><td></td><td>.26</td><td>.12</td><td>.48</td></tr><tr><td>Gabor Decomposition Steerable Pyramid</td><td>.24</td><td>.27 .32</td><td>.25</td></tr></table>
|
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| 314 |
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Figure 6: Images where predicted threshold is too high (”Overshoot”, where perturbation saliency is better than predicted) or too low (”Undershoot”), considered from several perceptual threshold ranges $\pm 2$ dB of shown number). Some images are reproduced from Figure 1.
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Figure 7: Background context for different DNN models (following figure 2).
|
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Figure 8: Background context for baseline DNN models (following figure 2). ”CaffeNet iter 310K” is reproduced from Figure 7.
|
| 322 |
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| 323 |
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Figure 9: Background context for Shape. Shown for each model is the measured MI for the six ”Hard” shapes as a function of the MI for the ”Easy” shape. The last panel shows an analagous comparison measured in human subjects by Weisstein & Harris (1974). A data point which lies below the dashed diagonal indicates a configuration for which discriminating line location is easier for the Easy shape compared with the relevant Hard shape.
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| 325 |
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Figure 10: Contrast sensitivity (following Figure 3) for DNN architectures CaffeNet, GoogLeNet, and ResNet-152.
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| 328 |
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Figure 11: Comparison of contrast sensitivity. Shown are iso-output curves, for which perceived contrast is the same (Human), or for which the $L _ { 1 }$ change relative to a gray image is the same (DNN model VGG-19). To obtain a correspondence between human frequency values (given in cycles per degree of visual field) to DNN frequency values (given in cycles per image), a scaling was chosen such that the minima of the blue curve is given at the same frequency value. Human data is for subject M.A.G. as measured by Georgeson & Sullivan (1975).
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| 331 |
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# 8 APPENDIX: EXPERIMENTAL SETUP
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| 333 |
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# 8.1 DNN MODELS
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| 335 |
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To collect DNN computation snapshots, we used MATLAB with MatConvNet version 1.0-beta20 (Vedaldi & Lenc, 2015). All MATLAB code will be made available upon acceptance of this manuscript. The pre-trained DNN models we have used are: CaffeNet (which is a variant of AlexNet provided in Caffe, Jia et al., 2014), GoogLeNet (Szegedy et al., 2014), VGG-19 (Simonyan & Zisserman, 2014), and ResNet-152 (He et al., 2015). The models were trained on the same ImageNet LSVRC. The CaffeNet model was trained using Caffe with the default ImageNet training parameters (stopping at iteration 310, 000) and imported into MatConvNet. For the GoogLeNet model, we used the imported pre-trained reference-Caffe implementation. For VGG-19 and ResNet-152, we used the imported pre-trained original versions. In all experiments input image size was $2 2 4 \times 2 2 4$ or $2 2 7 \times 2 2 7$ .
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# 8.2 BASELINE MODELS
|
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As baselines to compare with pre-trained DNN models, we consider: (a) a multiscale linear filter bank of Gabor functions, (b) a steerable-pyramid linear filter bank (Simoncelli & Freeman, 1995), (c) the VGG-19 model for which the learned parameters (weights) were randomly scrambled within layer, and (d) the CaffeNet model at multiple time points during training. For the Gabor decomposition, the following Gabor filters were used: all compositions of $\sigma = \bar { \{ 1 , 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \} } \mathrm { p x }$ , $\lambda = \{ 1 , 2 \} \cdot \sigma$ , orientation $\underline { { \underline { { \mathbf { \Pi } } } } } = \{ 0 , \pi / 3 , 2 \pi / 3 , \pi , 4 \pi / 3 , 5 \pi / 3 \}$ , and phase $= \{ 0 , \pi / 2 \}$ .
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# 8.3 IMAGE PERTURBATION EXPERIMENT
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The noiseless images were obtained from Alam et al. (2014). In main text, ”image scale” refers to percent coverage of DNN input. Since size of original images $( 1 4 9 \times 1 4 9 )$ is smaller than DNN input of $( 2 2 4 \times 2 2 4 )$ or $( 2 2 7 \times 2 2 7 )$ ), the images were resized by a factor of 1.5 so that $100 \%$ image scale covers approximately the entire DNN input area.
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Human psychophysics and DNN experiments were done for nearly identical images. A slight discrepancy relates to how the image is blended with the background in the special case where the region where noise is added has no image surround at one or two side. In these sides (which depend on the technical procedure with which images were obtained, see Alam et al., 2014), the surround blending here was hard, while the original was smooth.
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# 8.4 BACKGROUND CONTEXT EXPERIMENT
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# 8.4.1 SEGMENTATION
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The images used are based on the Texture Discrimination Task (Karni & Sagi, 1991). In the variant considered here (Pinchuk-Yacobi et al., 2015), subjects were presented with a grid of lines, all of which were horizontal, except two or three that were diagonal. Subjects discriminated whether the arrangement of diagonal lines is horizontal or vertical, and this discrimination was found to be more difficult when the central line is horizontal rather than diagonal (”Hard” vs. ”Easy” in Figure 2a). To limit human performance in this task, two manipulations were applied: (a) the location of each line in the pattern was jittered, and (b) a noise mask was presented briefly after the pattern. Here we only retained (a).
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A total of 90 configurations were tested, obtained by combinations of the following alternatives:
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• Three scales: line length of 9, 12.3, or $1 9 . 4 \ \mathrm { p x }$ (number of lines co-varied with line length, see Figure 12).
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• Three levels of location jittering, defined as a multiple of line length: $\{ 1 , 2 , 3 \} \cdot 0 . 0 6 2 5 \cdot l$ px, where $l$ is the length of a line in the pattern. Jittering was applied separately to each line in the pattern.
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• Ten locations of diagonal lines: center, random, four locations of half-distance from center to corners, four locations of half-distance from center to image borders.
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For each configuration, the discriminated arrangement of diagonal lines was either horizontal or vertical, and the central line was either horizontal or diagonal (i.e. hard or easy).
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Figure 12: Pattern scales used in the different configurations of the Segmentation condition. Actual images used were white-on-black rather than black-on-white.
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|
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+
# 8.4.2 CROWDING
|
| 366 |
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The images used are motivated by the crowding effect (Livne & Sagi, 2007; Pelli et al., 2004).
|
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| 369 |
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A total of 90 configurations were tested, obtained by combinations of the following alternatives:
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• Three scales: font size of 15.1, 20.6, or $3 2 . 4 { \mathrm { p x } }$ (see Figure 13).
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• Three levels of discriminated-letter location jittering, defined as a multiple of font size: $\{ 1 , 2 , 3 \} \cdot 0 . 0 6 2 5 \cdot l$ px, where $l$ is font size. The jitter of surround letters (M, N, S, and T) was fixed (i.e. the background was static).
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| 373 |
+
• Ten locations: center, random, four locations of half-distance from center to corners, four locations of half-distance from center to image borders.
|
| 374 |
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|
| 375 |
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For each configuration, the discriminated letter was either A, B, C, D, E, or F, and the background was either blank (easy) or composed of the letters M, N, S, and T (hard).
|
| 376 |
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|
| 377 |
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| 378 |
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Figure 13: Pattern scales used in the different configurations of the Crowding condition. Actual images used were white-on-black rather than black-on-white.
|
| 379 |
+
|
| 380 |
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# 8.4.3 SHAPE
|
| 381 |
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|
| 382 |
+
The images used are based on the object superiority effect by Weisstein & Harris (1974), where discriminating a line location is easier when combined with surrounding lines a shape is formed.
|
| 383 |
+
|
| 384 |
+
A total of 90 configurations were tested, obtained by combinations of the following alternatives:
|
| 385 |
+
|
| 386 |
+
• Three scales: discriminated-line length of 9, 15.1, or 22.7 px (see Figure 14). • Five levels of whole-pattern location jittering, defined as a multiple of discriminated-line length: $\{ 1 , 2 , 5 , 1 0 , 1 \bar { 5 } \} \cdot 0 . 0 6 2 5 \cdot l$ px, where $l$ is the length of the discriminated line.
|
| 387 |
+
|
| 388 |
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• Six ”hard” background line layouts (patterns $b { - } f$ of their Figure 2 and the additional pattern $f$ of their Figure 3 in Weisstein & Harris, 1974). The ”easy” layout was always the same (pattern $a$ ).
|
| 389 |
+
|
| 390 |
+
For each configuration, the line whose location is discriminated had four possible locations (two locations are shown in Figure 2c), and the surrounding background line layout could compose a shape (easy) or not (hard).
|
| 391 |
+
|
| 392 |
+
|
| 393 |
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|
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Figure 14: Pattern scales used in the different configurations of the Shape condition. Actual images used were white-on-black rather than black-on-white.
|
| 395 |
+
|
| 396 |
+
# 8.5 CONTRAST SENSITIVITY EXPERIMENT
|
| 397 |
+
|
| 398 |
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Used images depicted sine gratings at different contrast, spatial frequency, sine phase, and sine orientation combinations.
|
parse/train/BJbD_Pqlg/BJbD_Pqlg_content_list.json
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "HUMAN PERCEPTION IN COMPUTER VISION / CONFERENCE SUBMISSIONS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
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"bbox": [
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| 7 |
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| 9 |
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| 10 |
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| 11 |
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],
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| 12 |
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"page_idx": 0
|
| 13 |
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},
|
| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
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"text": "Ron Dekel ∗ ",
|
| 17 |
+
"text_level": 1,
|
| 18 |
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"bbox": [
|
| 19 |
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| 20 |
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| 21 |
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| 22 |
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| 23 |
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],
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| 24 |
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"page_idx": 0
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| 25 |
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},
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| 26 |
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{
|
| 27 |
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"type": "text",
|
| 28 |
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"text": "Department of Neurobiology Weizmann Institute of Science Rehovot, PA 7610001, Israel ron.dekel@weizmann.ac.il ",
|
| 29 |
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"bbox": [
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| 30 |
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| 31 |
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| 32 |
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| 33 |
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| 34 |
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],
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| 35 |
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"page_idx": 0
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| 36 |
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},
|
| 37 |
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{
|
| 38 |
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"type": "text",
|
| 39 |
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"text": "ABSTRACT ",
|
| 40 |
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"text_level": 1,
|
| 41 |
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"bbox": [
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| 42 |
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| 43 |
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| 44 |
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| 45 |
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| 46 |
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],
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| 47 |
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"page_idx": 0
|
| 48 |
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},
|
| 49 |
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{
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| 50 |
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"type": "text",
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| 51 |
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"text": "Computer vision has made remarkable progress in recent years. Deep neural network (DNN) models optimized to identify objects in images exhibit unprecedented task-trained accuracy and, remarkably, some generalization ability: new visual problems can now be solved more easily based on previous learning. Biological vision (learned in life and through evolution) is also accurate and generalpurpose. Is it possible that these different learning regimes converge to similar problem-dependent optimal computations? We therefore asked whether the human system-level computation of visual perception has DNN correlates and considered several anecdotal test cases. We found that perceptual sensitivity to image changes has DNN mid-computation correlates, while sensitivity to segmentation, crowding and shape has DNN end-computation correlates. Our results quantify the applicability of using DNN computation to estimate perceptual loss, and are consistent with the fascinating theoretical view that properties of human perception are a consequence of architecture-independent visual learning. ",
|
| 52 |
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"bbox": [
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| 53 |
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| 54 |
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| 55 |
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| 56 |
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| 58 |
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"page_idx": 0
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| 59 |
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},
|
| 60 |
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{
|
| 61 |
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"type": "text",
|
| 62 |
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"text": "1 QUICK EXPERT SUMMARY ",
|
| 63 |
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"text_level": 1,
|
| 64 |
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"bbox": [
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| 65 |
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| 66 |
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| 67 |
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| 68 |
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| 69 |
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| 70 |
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"page_idx": 0
|
| 71 |
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|
| 72 |
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{
|
| 73 |
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"type": "text",
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| 74 |
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"text": "Considering the learned computation of ImageNet-trained DNNs, we find: ",
|
| 75 |
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"type": "text",
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| 85 |
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"text": "• Large computation changes for perceptually salient image changes (Figure 1). \n• Gestalt: segmentation, crowding, and shape interactions in computation (Figure 2). \n• Contrast constancy: bandpass transduction in first layers is later corrected (Figure 3). ",
|
| 86 |
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],
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| 92 |
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| 93 |
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| 94 |
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{
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| 95 |
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"type": "text",
|
| 96 |
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"text": "These properties are reminiscent of human perception, perhaps because learned general-purpose classifiers (human and DNN) tend to converge. ",
|
| 97 |
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"bbox": [
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| 98 |
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| 104 |
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| 105 |
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{
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| 106 |
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"type": "text",
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| 107 |
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"text": "2 INTRODUCTION ",
|
| 108 |
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"text_level": 1,
|
| 109 |
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"bbox": [
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| 110 |
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| 111 |
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| 116 |
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},
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| 117 |
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{
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| 118 |
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"type": "text",
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| 119 |
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"text": "Deep neural networks (DNNs) are a class of computer learning algorithms that have become widely used in recent years (LeCun et al., 2015). By training with millions of examples, such models achieve unparalleled degrees of task-trained accuracy (Krizhevsky et al., 2012). This is not unprecedented on its own - steady progress has been made in computer vision for decades, and to some degree current designs are just scaled versions of long-known principles (Lecun et al., 1998). In previous models, however, only the design is general-purpose, while learning is mostly specific to the context of a trained task. Interestingly, for current DNNs trained to solve a large-scale image recognition problem (Russakovsky et al., 2014), the learned computation is useful as a building block for drastically different and untrained visual problems (Huh et al., 2016; Yosinski et al., 2014). ",
|
| 120 |
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| 121 |
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| 122 |
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| 123 |
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| 124 |
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| 125 |
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],
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| 126 |
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| 127 |
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| 128 |
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{
|
| 129 |
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"type": "text",
|
| 130 |
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"text": "For example, orientation- and frequency-selective features (Gabor patches) can be considered general-purpose visual computations. Such features are routinely discovered by DNNs (Krizhevsky et al., 2012; Zeiler & Fergus, 2013), by other learning algorithms (Hinton & Salakhutdinov, 2006; ",
|
| 131 |
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| 132 |
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| 133 |
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| 137 |
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| 138 |
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| 139 |
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{
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| 140 |
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"type": "text",
|
| 141 |
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"text": "Lee et al., 2008; 2009; Olshausen & Field, 1997), and are extensively hard-coded in computer vision (Jain & Farrokhnia, 1991). Furthermore, a similar computation is believed to underlie the spatial response properties of visual neurons of diverse animal phyla (Carandini et al., 2005; DeAngelis et al., 1995; Hubel & Wiesel, 1968; Seelig & Jayaraman, 2013), and is evident in human visual perception (Campbell & Robson, 1968; Fogel & Sagi, 1989; Neri et al., 1999). This diversity culminates in satisfying theoretical arguments as to why Gabor-like features are so useful in general-purpose vision (Olshausen, 1996; Olshausen & Field, 1997). ",
|
| 142 |
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"bbox": [
|
| 143 |
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| 144 |
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| 145 |
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| 147 |
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| 148 |
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"page_idx": 1
|
| 149 |
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|
| 150 |
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{
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| 151 |
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"type": "text",
|
| 152 |
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"text": "As an extension, general-purpose computations are perhaps of universal use. For example, a dimensionality reduction transformation that optimally preserves recognition-relevant information may constitute an ideal computation for both DNN and animal. More formally, different learning algorithms with different physical implementations may converge to the same computation when similar (or sufficiently general) problems are solved near-optimally. Following this line of reasoning, DNN models with good general-purpose computations may be computationally similar to biological visual systems, even more so than less accurate and less general biologically plausible simulations (Kriegeskorte, 2015; Yamins & DiCarlo, 2016). ",
|
| 153 |
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| 154 |
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| 155 |
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| 156 |
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| 158 |
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| 159 |
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|
| 160 |
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|
| 161 |
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| 162 |
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"type": "text",
|
| 163 |
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"text": "Related work seems to be consistent with computation convergence. First, different DNN training regimes seem to converge to a similar learned computation (Li et al., 2015; Zhou et al., 2014). Second, image representation may be similar in trained DNN and in biological visual systems. That is, when the same images are processed by DNN and by humans or monkeys, the final DNN computation stages are strong predictors of human fMRI and monkey electrophysiology data collected from visual areas V4 and IT (Cadieu et al., 2014; Khaligh-Razavi & Kriegeskorte, 2014; Yamins et al., 2014). Furthermore, more accurate DNN models exhibit stronger predictive power (Cadieu et al., 2014; Dubey & Agarwal, 2016; Yamins et al., 2014), and the final DNN computation stage is even a strong predictor of human-perceived shape discrimination (Kubilius et al., 2016). However, some caution is perhaps unavoidable, since measured similarity may be confounded with categorization consistency, view-invariance resilience, or similarity in the inherent difficulty of the tasks undergoing comparison. A complementary approach is to consider images that were produced by optimizing trained DNN-based perceptual metrics (Gatys et al., 2015a;b; Johnson et al., 2016; Ledig et al., 2016), which perhaps yields undeniable evidence of non-trivial computational similarity, although a more objective approach may be warranted. ",
|
| 164 |
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| 165 |
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| 170 |
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|
| 171 |
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},
|
| 172 |
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| 173 |
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"type": "text",
|
| 174 |
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"text": "Here, we quantify the similarity between human visual perception, as measured by psychophysical experiments, and individual computational stages (layers) in feed-forward DNNs trained on a large-scale image recognition problem (ImageNet LSVRC). Comparison is achieved by feeding the experimental image stimuli to the trained DNN and comparing a DNN metric (mean mutual information or mean absolute change) to perceptual data. The use of reduced (simplified and typically non-natural) stimuli ensures identical inherent task difficulty across compared categories and prevents confounding of categorization consistency with measured similarity. Perception, a systemlevel computation, may be influenced less by the architectural discrepancy (biology vs. DNN) than are neural recordings. ",
|
| 175 |
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| 183 |
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{
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| 184 |
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"type": "text",
|
| 185 |
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"text": "3 CORRELATE FOR IMAGE CHANGE SENSITIVITY ",
|
| 186 |
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"text_level": 1,
|
| 187 |
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"type": "text",
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| 197 |
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"text": "From a perceptual perspective, an image change of fixed size has different saliency depending on image context (Polat & Sagi, 1993). To investigate whether the computation in trained DNNs exhibits similar contextual modulation, we used the Local Image Masking Database (Alam et al., 2014), in which 1080 partially-overlapping images were subjected to different levels of the same random additive noise perturbation, and for each image, a psychophysical experiment determined the threshold noise level at which the added-noise image is discriminated from two noiseless copies at $7 5 \\%$ (Figure 1a). Threshold is the objective function that is compared with an $L _ { 1 }$ -distance correlate in the DNN representation. The scale of measured threshold was: ",
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"type": "equation",
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"img_path": "images/3d251d57138f12ed451d1c9b8935620a0035238ef35dba7a2afcd646a24de396.jpg",
|
| 209 |
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"text": "$$\n2 0 \\cdot \\log _ { 1 0 } \\left( \\frac { \\mathrm { s t d } \\left( n o i s e \\right) } { T } \\right) ,\n$$",
|
| 210 |
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"text_format": "latex",
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| 211 |
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},
|
| 219 |
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{
|
| 220 |
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"type": "text",
|
| 221 |
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"text": "where std $( n o i s e )$ is the standard deviation of the additive noise, and $T$ is the mean image pixel value calculated over the region where the noise is added (i.e. image center). ",
|
| 222 |
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"bbox": [
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"page_idx": 1
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},
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| 230 |
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{
|
| 231 |
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"type": "image",
|
| 232 |
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"img_path": "images/790ed25adca1327d42ec60be5b4b6dbf2d30a4be4b78b863a68199a6c956a87c.jpg",
|
| 233 |
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"image_caption": [
|
| 234 |
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"Figure 1: Predicting perturbation thresholds. a, For a fixed image perturbation, perceptual detection threshold (visualized by red arrow) depends on image context. b, Measured perceptual threshold is correlated with the average $L _ { 1 }$ change in DNN computation due to image perturbation (for DNN model VGG-19, image scale $= 1 0 0 \\%$ ). c, Explained variability $( R ^ { 2 } )$ of perceptual threshold data when $L _ { 1 }$ change is based on isolated computational layers for different input image scales. Same VGG-19 model as in (b). X-axis labels: data refers to raw image pixel data, $\\mathsf { c o n v ^ { * } } _ { - 1 }$ and ${ \\mathsf { f c } } _ { - } *$ are the before-ReLU output of a convolution and a fully-connected operation, respectively, and prob is the output class label probabilities vector. d, Example images for whcih predicted threshold in b is much higher than perceptually measured (”Overshoot”, where perturbation saliency is better than predicted), or vise versa (”Undershoot”). Examples are considered from several perceptual threshold ranges $\\pm 2 \\mathrm { d B }$ of shown number). "
|
| 235 |
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],
|
| 236 |
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"image_footnote": [],
|
| 237 |
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"bbox": [
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| 238 |
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| 239 |
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| 240 |
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| 241 |
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| 242 |
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| 243 |
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"page_idx": 2
|
| 244 |
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},
|
| 245 |
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{
|
| 246 |
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"type": "text",
|
| 247 |
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"text": "The DNN correlate of perceptual threshold we used was the average $L _ { 1 }$ change in DNN computation between added-noise images and the original, noiseless image. Formally, ",
|
| 248 |
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"bbox": [
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| 249 |
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},
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| 256 |
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{
|
| 257 |
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"type": "equation",
|
| 258 |
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"img_path": "images/a5b5cf26a3b0d9b2e0b128000c5a76eeaf41b7c2988723a760776a8a40d73506.jpg",
|
| 259 |
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"text": "$$\nL _ { 1 } ^ { i , n } ( I ) = \\left| \\overline { { { a _ { i } \\left( I + n o i s e \\left( n \\right) \\right) } } } - a _ { i } \\left( I \\right) \\right| ,\n$$",
|
| 260 |
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"text_format": "latex",
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| 261 |
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"bbox": [
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| 265 |
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| 267 |
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"page_idx": 2
|
| 268 |
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},
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| 269 |
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{
|
| 270 |
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"type": "text",
|
| 271 |
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"text": "where $a _ { i } \\left( X \\right)$ is the activation value of neuron $i$ during the DNN feedforward pass for input image $X$ , and the inner average (denoted by bar) is taken over repetitions with random $n$ -sized noise (noise is introduced at random phase spectra in a fixed image location, an augmentation that follows the between-image randomization described by Alam et al., 2014; the number of repetitions was 10 or more). Unless otherwise specified, the final $L _ { 1 }$ prediction is $L _ { 1 } ^ { i , n }$ averaged across noise levels $( - 4 0$ to $2 5 { \\mathrm { ~ d B } }$ with 5-dB intervals) and computational neurons (first within and then across computational stages). Using $L _ { 1 }$ averaged across noise levels as a correlate for the noise level of perceptual threshold is a simple approximation with minimal assumptions. ",
|
| 272 |
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},
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| 281 |
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"type": "text",
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| 282 |
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"text": "Results show that the $L _ { 1 }$ metric is correlated with the perceptual threshold for all tested DNN architectures (Figure 1b, 4a-c). In other words, higher values of the $L _ { 1 }$ metric (indicating larger changes in DNN computation due to image perturbation, consistent with higher perturbation saliency) are correlated with lower values of measured perceptual threshold (indicating that weaker noise levels are detectable, i.e. higher saliency once more). ",
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| 283 |
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"bbox": [
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{
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| 292 |
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"type": "table",
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"img_path": "images/6f25fd6a861f747a1e0077f8db84fa394784c42bcca979c0fae7ffd06541dec3.jpg",
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| 294 |
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"table_caption": [
|
| 295 |
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"Table 1: Prediction accuracy. Percent of linearly explained variability $( R ^ { 2 } )$ , absolute value of Spearman rank-order correlation coefficient (SROCC), and the root mean squared error of the linear prediction (RMSE) are presented for each prediction model. Note the measurement scale of the threshold data being predicted (Eq. 1). $( ^ { * } )$ Thresholds linearized through a logistic transform before prediction (see Larson & Chandler, 2010), possibly increasing but not decreasing measured predictive strength. $( ^ { * * } )$ Average of four similar alternatives. "
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| 296 |
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],
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"table_footnote": [],
|
| 298 |
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"table_body": "<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td></tr><tr><td>Signal-noise ratio</td><td></td><td></td><td></td></tr><tr><td>Spectral change</td><td>.20 .25</td><td>.39</td><td>7.67</td></tr><tr><td>RMS contrast (Alam et al., 2014)</td><td>.27*</td><td>.61</td><td>7.42</td></tr><tr><td>L1 VGG-19 (50%)</td><td>.57</td><td>.46</td><td>1</td></tr><tr><td>L1 VGG-19 (66%)</td><td>.60</td><td>.77</td><td>5.57 5.42</td></tr><tr><td>L1 VGG-19 (100%)</td><td>.60</td><td>.79</td><td></td></tr><tr><td>Perceptual model** (Alam et al., 2014)</td><td>.60*</td><td>.79</td><td>5.40</td></tr><tr><td></td><td></td><td>.70</td><td>5.73</td></tr><tr><td>Inter-person (Alam et al., 2014)</td><td>.84*</td><td>.87</td><td>4.08</td></tr></table>",
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"type": "text",
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"text": "",
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"type": "text",
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| 320 |
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"text": "To quantify and compare predictive power, we considered the percent of linearly explained variability $( R ^ { 2 } )$ . For all tested DNN architectures, the prediction explains about $6 0 \\%$ of the perceptual variability (Tables 1, 2; baselines at Tables 3-5), where inter-person similarity representing theoretical maximum is $84 \\%$ (Alam et al., 2014). The DNN prediction is far more accurate than a prediction based on simple image statistical properties (e.g. RMS contrast), and is on par with a detailed perceptual model that relies on dozens of psychophysically collected parameters (Alam et al., 2014). The Spearmann correlation coefficient is much higher compared with the perceptual model (with an absolute SROCC value of about 0.79 compared with 0.70, Table 1), suggesting that the $L _ { 1 }$ metric gets the order right but not the scale. We did not compare these results with models that fit the experimental data (e.g. Alam et al., 2015; Liu & Allebach, 2016), since the $L _ { 1 }$ metric has no explicit parameters. Also, different DNN architectures exhibited high similarity in their predictions $\\bar { R } ^ { 2 }$ of about 0.9, e.g. Figure 4d). ",
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"type": "text",
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| 331 |
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"text": "Prediction can also be made from isolated computational stages, instead of across all stages as before. This analysis shows that the predictive power peaks mid-computation across all tested image scales (Figure 1c). This peak is consistent with use of middle DNN layers to optimize perceptual metrics (Gatys et al., 2015a;b; Ledig et al., 2016), and is reminiscent of cases in which low- to mid-level vision is the performance limiting computation in the detection of at-threshold stimuli (Campbell & Robson, 1968; Del Cul et al., 2007). ",
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"type": "text",
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"text": "Finally, considering the images for which the $L _ { 1 }$ -based prediction has a high error suggests a factor which causes a systematic inconsistency with perception (Figures 1d, 6). This factor may be related to the mean image luminance: by introducing noise perturbations according to the scale of Equation 1, a fixed noise size (in dB) corresponds to smaller pixel changes in dark compared with bright images. (Using this scales reflects an assumption of multiplicative rather than additive conservation; this assumption may be justified for the representation at the final but perhaps not the intermediate computational stages considering the log-linear contrast response discussed in Section 5). Another factor may the degree to which image content is identifiable. ",
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"type": "text",
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| 353 |
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"text": "The previous analysis suggested gross computational similarity between human perception and trained DNNs. Next, we aimed to extend the comparison to more interpretable properties of perception by considering more highly controlled designs. To this end, we considered cases in which a static background context modulates the difficulty of discriminating a foreground shape, despite no spatial overlap of foreground and background. This permits interpretation by considering the cause of the modulation. ",
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"type": "text",
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| 364 |
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"text": "We first consider segmentation, in which arrangement is better discriminated for arrays of consistently oriented lines compared with inconsistently oriented lines (Figure 2a) (Pinchuk-Yacobi et al., 2016). Crowding is considered next, where surround clutter that is similar to the discriminated target leads to deteriorated discrimination performance (Figure 2b) (Livne & Sagi, 2007). Last to be addressed is object superiority, in which a target line location is better discriminated when it is in a shape-forming layout (Figure 2c) (Weisstein & Harris, 1974). In this case, clutter is controlled by having the same fixed number of lines in context. To measure perceptual discrimination, these works introduced performance-limiting manipulations such as location jittering, brief presentation, and temporal masking. While different manipulations showed different measured values, order-of-difficulty was typically preserved. Here we changed all the original performance-limiting manipulations to location jittering (whole-shape or element-wise, see Section 8.4). ",
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"type": "text",
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"text": "To quantify discrimination difficulty in DNNs, we measured the target-discriminative information of isolated neurons (where performance is limited by location jittering noise), then averaged across all neurons (first within and then across computational layer stages). Specifically, for each neuron, we measured the reduction in categorization uncertainty due to observation, termed mutual information (MI): ",
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"type": "equation",
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"img_path": "images/895dc524952cb1910c247a031be03445427fd8959bb40023011e072fa6e6a1d7.jpg",
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| 387 |
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"text": "$$\nM I \\left( A _ { i } ; C \\right) = H \\left( C \\right) - H \\left( C \\vert A _ { i } \\right) ,\n$$",
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| 388 |
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"text_format": "latex",
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| 389 |
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"bbox": [
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{
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"type": "text",
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| 399 |
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"text": "where H stands for entropy, and $A _ { i }$ is a random variable for the value of neuron i when the DNN processes a random image from a category defined by the random variable C. For example, if a neuron gives a value in the range of 100.0 to 200.0 when the DNN processes images from category A, and 300.0 to 400.0 for category B, then the category is always known by observing the value, and so mutual information is high $\\mathbf { M } \\mathbf { I } { = } 1$ bits). On the other extreme, if the neuron has no discriminative task information, then ${ \\bf M I } { = } 0$ bits. To measure MI, we quantized activations into eight equal-amount bins, and used 500 samples (repetitions having different location jittering noise) across categories. The motivation for this correlate is the assumption that the perceptual order-of-difficulty reflects the quantity of task-discriminative information in the representation. ",
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"type": "text",
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"text": "Results show that, across hundreds of configurations (varying pattern element size, target location, jitter magnitude, and DNN architecture; see Section 8.4), the qualitative order of difficulty in terms of the DNN MI metric is consistent with the order of difficulty measured in human psychophysical experiments, for the conditions addressing segmentation and crowding (Figures 2d, 7; for baseline models see Figure 8). It is interesting to note that the increase in similarity develops gradually along different layer types in the DNN computation (i.e. not just pooling layers), and is accompanied by a gradual increase in the quantity of task-relevant information (Figure 2e-g). This indicates a link between task relevance and computational similarity for the tested conditions. Note that unlike the evident increase in isolated unit task information, the task information from all units combined decreases by definition along any computational hierarchy. An intuition for this result is that the total hidden information decreases, while more accessible per-unit information increases. ",
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| 411 |
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"bbox": [
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"type": "text",
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"text": "For shape formation, four out of six shapes consistently show order of difficulty like perception, and two shapes consistently do no (caricature at Figure 2h; actual data at Figure 9). ",
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"bbox": [
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{
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| 431 |
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"type": "image",
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| 432 |
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"img_path": "images/facc45e6917a8622804e2ee023233bcc6bd542f3ced7648ea2c3f55e2650d3af.jpg",
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"image_caption": [
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| 434 |
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"Figure 2: Background context. a-c, Illustrations of reproduced discrimination stimuli for three psychophysical experiments (actual images used were white-on-black rather than black-on-white, and pattern size was smaller, see Figures 12-14). d, Number of configurations for which orderof-difficulty in discrimination is qualitatively consistency with perception according to a mutual information DNN metric. Configurations vary in pattern (element size, target location, and jitter magnitude; see Section 8.4) and in DNN architecture used (CaffeNet, GoogLeNet, VGG-19, and ResNet-152). DNN metric is the average across neurons of the isolated neuron target-discriminative information (averaged first within, and then across computational layer stages), where performance is limited by location jittering (e.g. evident jitter in illustrations). e-g, The value of the MI metric across computational layers of model VGG-19 for a typical pattern configuration. The six ”hard” (gray) lines in Shape MI correspond to six different layouts (see Section 8.4.3). Analysis shows that for isolated computation stages, similarity to perception is evident only at the final DNN computation stages. h, A caricature summarizing the similarity and discrepancy of perception and the MI-based DNN prediction for Shape (see Figure 9). "
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| 435 |
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],
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| 436 |
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},
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{
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| 446 |
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"type": "text",
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"text": "5 CORRELATE FOR CONTRAST SENSITIVITY ",
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| 448 |
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"text_level": 1,
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| 458 |
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"type": "text",
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| 459 |
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"text": "A cornerstone of biological vision research is the use of sine gratings at different frequencies, orientations, and contrasts (Campbell & Robson, 1968). Notable are results showing that the lowest perceivable contrast in human perception depends on frequency. Specifically, high spatial frequencies are attenuated by the optics of the eye, and low spatial frequencies are believed to be attenuated due to processing inefficiencies (Watson & Ahumada, 2008), so that the lowest perceivable contrast is found at intermediate frequencies. (To appreciate this yourself, examine Figure 3a). Thus, for low-contrast gratings, the physical quantity of contrast is not perceived correctly: it is not preserved across spatial frequencies. Interestingly, this is corrected for gratings of higher contrasts, for which perceived contrast is more constant across spatial frequencies (Georgeson & Sullivan, 1975). ",
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{
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| 469 |
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"type": "text",
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| 470 |
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"text": "The DNN correlate we considered is the mean absolute change in DNN representation between a gray image and sinusoidal gratings, at all combinations of spatial frequency and contrast. Formally, for neurons in a given layer, we measured: ",
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| 471 |
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"type": "equation",
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"img_path": "images/e06538e1f986e9ff0de91003d39525212816001f502e80a4bc3b1aaab836db9a.jpg",
|
| 482 |
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"text": "$$\nL _ { 1 } ( c o n t r a s t , f r e q u e n c y ) = \\frac { 1 } { N _ { n e u r o n s } } \\sum _ { i = 1 } ^ { N _ { n e u r o n s } } \\left| \\overline { { a _ { i } \\left( c o n t r a s t , f r e q u e n c y \\right) } } - a _ { i } \\left( 0 , 0 \\right) \\right| ,\n$$",
|
| 483 |
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"text_format": "latex",
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"bbox": [
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},
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| 492 |
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{
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| 493 |
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"type": "text",
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| 494 |
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"text": "where $a _ { i }$ (contrast, frequency) is the average activation value of neuron $i$ to 250 sine images (random orientation, random phase), $a _ { i } \\left( 0 , 0 \\right)$ is the response to a blank (gray) image, and $N _ { n }$ eurons is the number of neurons in the layer. This measure reflects the overall change in response vs. the gray image. ",
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"type": "text",
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"text": "Results show a bandpass response for low-contrast gratings (blue lines strongly modulated by frequency, Figures 3, 10), and what appears to be a mostly constant response at high contrast for end-computation layers (red lines appear more invariant to frequency), in accordance with perception. ",
|
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"type": "text",
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"text": "We next aimed to compare these results with perception. Data from human experiments is generally iso-output (i.e. for a pre-set output, such as $7 5 \\%$ detection accuracy, the input is varied to find the value which produce the preset output). However, the DNN measurements here are iso-input (i.e. for a fixed input contrast the $L _ { 1 }$ is measured). As such, human data should be compared to the interpoalted inverse of DNN measurements. Specifically, for a set output value, the interpolated contrast value which produce the output is found for every frequency (Figure 11). This analysis permits quantifying the similarity of iso-output curves for human and DNN, measured here as the percent of log-Contrast variability in human measurements which is explained by the DNN predictions. This showed a high explained variability at the end computation stage (prob layer, $\\mathbf { \\dot { \\mathit { R } } ^ { 2 } = 9 4 \\% }$ , but importantly, a similarly high value at the first computational stage (conv1 1 layer, $R ^ { 2 } = 9 6 \\%$ ). Intiutively, while the ”internal representation” variability in terms of $L _ { 1 }$ is small, the iso-output number-of-input-contrast-cahnges variability is still high. For example. for the prob layer, about the same $L _ { 1 }$ is measured for (Contrast $^ { - 1 }$ ,freq $= 7 5$ ) and for (Contras ${ = } 0 . 1 8$ ,freq $= 1 2$ ). ",
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| 517 |
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| 524 |
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| 525 |
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| 526 |
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"type": "text",
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| 527 |
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"text": "An interesting, unexpected observation is that the logarithmically spaced contrast inputs are linearly spaced at the end-computation layers. That is, the average change in DNN representation scales logarithmically with the size of input change. This can be quantified by the correlation of output $L _ { 1 }$ with log Contrast input, which showed $R ^ { 2 } = 9 8 \\%$ (averaged across spatial frequencies) for prob, while much lower values were observed for early and middle layers (up to layer fc7). The same computation when scrambling the learned parameters of the model showed $R ^ { 2 } = 6 0 \\%$ . Because the degree of log-linearity observed was extremely high, it may be an important emergent property of the learned DNN computation, which may deserve further investigation. However, this property is only reminiscent and not immediately consistent with the perceptual power-law scaling (Gottesman et al., 1981). ",
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| 528 |
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},
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| 536 |
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{
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| 537 |
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"type": "image",
|
| 538 |
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"img_path": "images/129e93025def5a9cbbe9b8278b8c41324aea4c3845d014896910bb2486ca54c3.jpg",
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| 539 |
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"image_caption": [
|
| 540 |
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"Figure 3: Contrast sensitivity. a. Perceived contrast is strongly affected by spatial frequency at low contrast, but less so at high contrast (which preserves the physical quantity of contrast and thus termed constancy). b. The $L _ { 1 }$ change in VGG-19 representation between a gray image and images depicting sinusoidal gratings at each combination of sine spatial frequency $\\mathbf { \\dot { x } }$ -axis) and contrast (color) (random orientation, random phase), considering the raw image pixel data representation (data), the before-ReLU output of the first convolutional layer representation (conv1 1), the output of the last fully-connected layer representation (fc8), and the output class label probabilities representation (prob). "
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| 541 |
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],
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| 542 |
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},
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| 551 |
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{
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| 552 |
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"type": "text",
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| 553 |
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"text": "6 DISCUSSION ",
|
| 554 |
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{
|
| 564 |
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"type": "text",
|
| 565 |
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"text": "6.1 HUMAN PERCEPTION IN COMPUTER VISION ",
|
| 566 |
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"text_level": 1,
|
| 567 |
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"bbox": [
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| 568 |
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| 576 |
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"type": "text",
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| 577 |
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"text": "It may be tempting to believe that what we see is the result of a simple transformation of visual input. Centuries of psychophysics have, however, revealed complex properties in perception, by crafting stimuli that isolate different perceptual properties. In our study, we used the same stimuli to investigate the learned properties of deep neural networks (DNNs), which are the leading computer vision algorithms to date (LeCun et al., 2015). ",
|
| 578 |
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"bbox": [
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| 580 |
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| 585 |
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| 586 |
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|
| 587 |
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"type": "text",
|
| 588 |
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"text": "The DNNs we used were trained in a supervised fashion to assign labels to input images. To some degree, this task resembles the simple verbal explanations given to children by their parents. Since human perception is obviously much richer than the simple external supervision provided, we were not surprised to find that the best correlate for perceptual saliency of image changes is a part of the DNN computation that is only supervised indirectly (i.e. the mid-computation stage). This similarity is so strong, that even with no fine-tuning to human perception, the DNN metric is competitively accurate, even compared with a direct model of perception. ",
|
| 589 |
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"bbox": [
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| 597 |
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|
| 598 |
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"type": "text",
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| 599 |
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"text": "This strong, quantifiable similarity to a gross aspect of perception may, however, reflect a mix of similarities and discrepancies in different perceptual properties. To address isolated perceptual effects, we considered experiments that manipulate a spatial interaction, where the difficulty of discriminating a foreground target is modulated by a background context. Results showed modulation of DNN target diagnostic, isolated unit information, consistent with the modulation found in perceptual discrimination. This was shown for contextual interactions reflecting grouping/segmentation (Harris et al., 2015), crowding/clutter (Livne & Sagi, 2007; Pelli et al., 2004), and shape superiority (Weisstein & Harris, 1974). DNN similarity to these groupings/gestalt phenomena appeared at the end-computation stages. ",
|
| 600 |
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"bbox": [
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| 602 |
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],
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| 606 |
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"page_idx": 7
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| 608 |
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{
|
| 609 |
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"type": "text",
|
| 610 |
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"text": "No less interesting, are the cases in which there is no similarity. For example, perceptual effects related to 3D (Erdogan & Jacobs, 2016) and symmetry (Pramod & Arun, 2016) do not appear to have a strong correlate in the DNN computation. Indeed, it may be interesting to investigate the influence of visual experience in these cases. And, equally important, similarity should be considered in terms of specific perceptual properties rather than as a general statement. ",
|
| 611 |
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"bbox": [
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"page_idx": 7
|
| 618 |
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},
|
| 619 |
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{
|
| 620 |
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"type": "text",
|
| 621 |
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"text": "6.2 RECURRENT VS. FEEDFORWARD CONNECTIVITY",
|
| 622 |
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"text_level": 1,
|
| 623 |
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"bbox": [
|
| 624 |
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|
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|
| 630 |
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},
|
| 631 |
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{
|
| 632 |
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"type": "text",
|
| 633 |
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"text": "In the human hierarchy of visual processing areas, information is believed to be processed in a feedforward sweep, followed by recurrent processing loops (top-down and lateral) (Lamme & Roelfsema, 2000). Thus, for example, the early visual areas can perform deep computations. Since mapping from visual areas to DNN computational layers is not simple, it will not be considered here. (Note that ResNet connectivity is perhaps reminiscent of unrolled recurrent processing). ",
|
| 634 |
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"bbox": [
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| 635 |
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| 636 |
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|
| 639 |
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],
|
| 640 |
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"page_idx": 7
|
| 641 |
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},
|
| 642 |
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{
|
| 643 |
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"type": "text",
|
| 644 |
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"text": "Interestingly, debate is ongoing about the degree to which visual perception is dependent on recurrent connectivity (Fabre-Thorpe et al., 1998; Hung et al., 2005): recurrent representations are obviously richer, but feedforward computations converge much faster. An implicit question here regarding the extent of feasible feed-forward representations is, perhaps: Can contour segmentation, contextual influences, and complex shapes be learned? Based on the results reported here for feedforward DNNs, a feedforward representation may seem sufficient. However, the extent to which this is true may be very limited. In this study we used small images with a small number of lines, while effects such as contour integration seem to take place even in very large configurations (Field et al., 1993). Such scaling seems more likely in a recurrent implementation. As such, a reasonable hypothesis may be that the full extent of contextual influence is only realizable with recurrence, while feedforward DNNs learn a limited version by converging towards a useful computation. ",
|
| 645 |
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"bbox": [
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| 652 |
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},
|
| 653 |
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{
|
| 654 |
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"type": "text",
|
| 655 |
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"text": "6.3 IMPLICATIONS AND FUTURE WORK ",
|
| 656 |
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"text_level": 1,
|
| 657 |
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"bbox": [
|
| 658 |
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"page_idx": 7
|
| 664 |
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},
|
| 665 |
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{
|
| 666 |
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"type": "text",
|
| 667 |
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"text": "6.3.1 USE IN BRAIN MODELING ",
|
| 668 |
+
"text_level": 1,
|
| 669 |
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"bbox": [
|
| 670 |
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| 671 |
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| 672 |
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| 673 |
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|
| 674 |
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],
|
| 675 |
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"page_idx": 7
|
| 676 |
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},
|
| 677 |
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{
|
| 678 |
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"type": "text",
|
| 679 |
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"text": "The use of DNNs in modeling of visual perception (or of biological visual systems in general) is subject to a tradeoff between accuracy and biological plausibility. In terms of architecture, other deep models better approximate our current understanding of the visual system (Riesenhuber & ",
|
| 680 |
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"bbox": [
|
| 681 |
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174,
|
| 682 |
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| 683 |
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825,
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| 684 |
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946
|
| 685 |
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],
|
| 686 |
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"page_idx": 7
|
| 687 |
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},
|
| 688 |
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{
|
| 689 |
+
"type": "text",
|
| 690 |
+
"text": "Poggio, 1999; Serre, 2014). However, the computation in trained DNN models is quite generalpurpose (Huh et al., 2016; Yosinski et al., 2014) and offers unparalleled accuracy in recognition tasks (LeCun et al., 2015). Since visual computations are, to some degree, task- rather than architecturedependent, an accurate and general-purpose DNN model may better resemble biological processing than less accurate biologically plausible ones (Kriegeskorte, 2015; Yamins & DiCarlo, 2016). We support this view by considering a controlled condition in which similarity is not confounded with task difficulty or categorization consistency. ",
|
| 691 |
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"bbox": [
|
| 692 |
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| 693 |
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|
| 697 |
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"page_idx": 8
|
| 698 |
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},
|
| 699 |
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{
|
| 700 |
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"type": "text",
|
| 701 |
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"text": "6.3.2 USE IN PSYCHOPHYSICS ",
|
| 702 |
+
"text_level": 1,
|
| 703 |
+
"bbox": [
|
| 704 |
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176,
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| 705 |
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241,
|
| 706 |
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397,
|
| 707 |
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256
|
| 708 |
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],
|
| 709 |
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"page_idx": 8
|
| 710 |
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},
|
| 711 |
+
{
|
| 712 |
+
"type": "text",
|
| 713 |
+
"text": "Our results imply that trained DNN models have good predictive value for outcomes of psychophysical experiments, permitting a zero-cost first-order approximation. Note, however, that the scope of such simulations may be limited, since learning (Sagi, 2011) and adaptation (Webster, 2011) were not considered here. ",
|
| 714 |
+
"bbox": [
|
| 715 |
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174,
|
| 716 |
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266,
|
| 717 |
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825,
|
| 718 |
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321
|
| 719 |
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],
|
| 720 |
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"page_idx": 8
|
| 721 |
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},
|
| 722 |
+
{
|
| 723 |
+
"type": "text",
|
| 724 |
+
"text": "Another fascinating option is the formation of hypotheses in terms of mathematically differentiable trained-DNN constraints, whereby it is possible to efficiently solve for the visual stimuli that optimally dissociate the hypotheses (see Gatys et al. 2015a;b; Mordvintsev et al. 2015 and note Goodfellow et al. 2014; Szegedy et al. 2013). The conclusions drawn from such stimuli can be independent of the theoretical assumptions about the generating process (for example, creating new visual illusions that can be seen regardless of how they were created). ",
|
| 725 |
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"bbox": [
|
| 726 |
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174,
|
| 727 |
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329,
|
| 728 |
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|
| 729 |
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412
|
| 730 |
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],
|
| 731 |
+
"page_idx": 8
|
| 732 |
+
},
|
| 733 |
+
{
|
| 734 |
+
"type": "text",
|
| 735 |
+
"text": "6.3.3 USE IN ENGINEERING (A PERCEPTUAL LOSS METRIC) ",
|
| 736 |
+
"text_level": 1,
|
| 737 |
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"bbox": [
|
| 738 |
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|
| 739 |
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|
| 740 |
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598,
|
| 741 |
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444
|
| 742 |
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],
|
| 743 |
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"page_idx": 8
|
| 744 |
+
},
|
| 745 |
+
{
|
| 746 |
+
"type": "text",
|
| 747 |
+
"text": "As proposed previously (Dosovitskiy & Brox, 2016; Johnson et al., 2016; Ledig et al., 2016), the saliency of small image changes can be estimated as the representational distance in trained DNNs. Here, we quantified this approach by relying on data from a controlled psychophysical experiment (Alam et al., 2014). We found the metric to be far superior to simple image statistical properties, and on par with a detailed perceptual model (Alam et al., 2014). This metric can be useful in image compression, whereby optimizing degradation across image sub-patches by comparing perceptual loss may minimize visual artifacts and content loss. ",
|
| 748 |
+
"bbox": [
|
| 749 |
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174,
|
| 750 |
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454,
|
| 751 |
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825,
|
| 752 |
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553
|
| 753 |
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],
|
| 754 |
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"page_idx": 8
|
| 755 |
+
},
|
| 756 |
+
{
|
| 757 |
+
"type": "text",
|
| 758 |
+
"text": "ACKNOWLEDGMENTS ",
|
| 759 |
+
"text_level": 1,
|
| 760 |
+
"bbox": [
|
| 761 |
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176,
|
| 762 |
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|
| 763 |
+
326,
|
| 764 |
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583
|
| 765 |
+
],
|
| 766 |
+
"page_idx": 8
|
| 767 |
+
},
|
| 768 |
+
{
|
| 769 |
+
"type": "text",
|
| 770 |
+
"text": "We thank Yoram Bonneh for his valuable questions which led to much of this work. ",
|
| 771 |
+
"bbox": [
|
| 772 |
+
176,
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| 773 |
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594,
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],
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"page_idx": 8
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| 778 |
+
},
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| 779 |
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{
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| 780 |
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"type": "text",
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| 781 |
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"text": "REFERENCES ",
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"Figure 4: Predicting perceptual sensitivity to image changes (following Figure 1). a-c, The $L _ { 1 }$ change in CaffeNet, GoogLeNet, and ResNet-152 DNN architectures as a function of perceptual threshold. d, The $L _ { 1 }$ change in GoogLeNet as a function of the $L _ { 1 }$ change in VGG-19. "
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"Figure 5: Prediction accuracy as a function of computational stage. a, Predicting perceptual sensitivity for model VGG-19 using the best single kernel (i.e. using one fitting parameter, no cross validation), vs. the standard $L _ { 1 }$ metric (reproduced from Figure 1). b, For non-branch computational stages of model ResNet-152. "
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"img_path": "images/6479245d1c7064bc0c96164be8f4cb7c4e24c26fe29e0a6de01f4565cc15533c.jpg",
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"table_caption": [
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"Table 2: Accuracy of perceptual sensitivity prediction and task-trained ImageNet center-crop top-1 validation accuracy for different DNN models (following Table 1 from which third row is reproduced; used scale: $100 \\%$ ). The quality of prediction for ResNet-152 improves dramatically if only the first tens of layers are considered (see Figure 5b). "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td><td>Recognition accuracy</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CaffeNet</td><td>.59</td><td>.78</td><td>5.44</td><td>56%</td></tr><tr><td>GoogLeNet</td><td>.59</td><td>.79</td><td>5.45 5.40</td><td>66%</td></tr><tr><td>VGG-19</td><td>.60</td><td>.79</td><td>5.82</td><td>70%</td></tr><tr><td>ResNet-152</td><td>.53</td><td>.74</td><td></td><td>75%</td></tr></table>",
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"bbox": [
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"img_path": "images/92528d6df66d5d2c95ed09ea268b2c7162af0848aa4b03967c2a149c25f461c1.jpg",
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"Table 3: Accuracy of perceptual sensitivity prediction for baseline models (see Section 8.2; used scale: $100 \\%$ ). "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>VGG-19, scrambled weights</td><td>.18</td><td>.39</td><td>7.76</td></tr><tr><td>Gabor filter bank</td><td>.32</td><td>.12</td><td>8.03</td></tr><tr><td>Steerable-pyramid filter bank</td><td>.37</td><td>.15</td><td>7.91</td></tr></table>",
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"bbox": [
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300,
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207
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"page_idx": 14
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{
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"type": "table",
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"img_path": "images/d299515359778da9a39093403eb9f20752e414a8701343cd5c2a594de36cd292.jpg",
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"table_caption": [
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"Table 4: Accuracy of perceptual sensitivity prediction during CaffeNet model standard training (used scale: $100 \\%$ ). Last row reproduced from Table 2. "
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],
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"table_footnote": [],
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"table_body": "<table><tr><td>Model</td><td>R²</td><td>SROCC</td><td>RMSE</td><td>Recognition accuracy</td></tr><tr><td>CaffeNetiter1</td><td></td><td></td><td>6.30</td><td></td></tr><tr><td>CaffeNetiter50K</td><td>.46 .59</td><td>.67 .79</td><td>5.43</td><td>0% 37%</td></tr><tr><td>CaffeNetiter100K</td><td>.60</td><td>.79</td><td>5.41</td><td>39%</td></tr><tr><td>CaffeNetiter150K</td><td>.60</td><td>.78</td><td>5.43</td><td>53%</td></tr><tr><td>CaffeNetiter200K</td><td>.59</td><td>.78</td><td>5.45</td><td>54%</td></tr><tr><td>CaffeNetiter250K</td><td>.59</td><td>.78</td><td>5.43</td><td>56%</td></tr><tr><td>CaffeNetiter300K</td><td>.59</td><td>.78</td><td>5.44</td><td>56%</td></tr><tr><td>CaffeNetiter310K</td><td>.59</td><td>.78</td><td>5.44</td><td>56%</td></tr></table>",
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| 1630 |
+
"img_path": "images/6fd105d55153a1a933f45a567507588fed95d08ba89bbe3a5aee32895d4cc78f.jpg",
|
| 1631 |
+
"table_caption": [
|
| 1632 |
+
"Table 5: Robustness of perceptual sensitivity prediction for varying prediction parameters for model VGG-19. First three rows reproduced from Table 1. Measurements for the lower noise range of -60:-40 dB were omitted by mistake. "
|
| 1633 |
+
],
|
| 1634 |
+
"table_footnote": [],
|
| 1635 |
+
"table_body": "<table><tr><td>Scale</td><td>Metric</td><td>Augmentation</td><td>Noise range</td><td>R²</td><td>SROCC</td><td>RMSE</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>100%</td><td>L1</td><td>noise phase</td><td>-40:25 dB</td><td>.60</td><td>.79</td><td>5.40</td></tr><tr><td>66%</td><td>L1</td><td>noise phase</td><td>-40:25 dB</td><td>.60</td><td>.79</td><td>5.42</td></tr><tr><td>50%</td><td>L1</td><td>noise phase</td><td>-40:25 dB</td><td>.57</td><td>.77</td><td>5.57</td></tr><tr><td>100%</td><td>L2</td><td>noise phase</td><td>-40:25 dB</td><td>.62</td><td>.80</td><td>5.29</td></tr><tr><td>100%</td><td>L1</td><td>None</td><td>-40:25 dB</td><td>.58</td><td>.77</td><td>5.55</td></tr><tr><td>100%</td><td>L1</td><td>noise phase</td><td>-20:25 dB</td><td>.59</td><td>.78</td><td>5.46</td></tr><tr><td>100%</td><td>L1</td><td>noise phase</td><td>-40:5 dB</td><td>.59</td><td>.79</td><td>5.43</td></tr></table>",
|
| 1636 |
+
"bbox": [
|
| 1637 |
+
232,
|
| 1638 |
+
489,
|
| 1639 |
+
764,
|
| 1640 |
+
621
|
| 1641 |
+
],
|
| 1642 |
+
"page_idx": 14
|
| 1643 |
+
},
|
| 1644 |
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{
|
| 1645 |
+
"type": "table",
|
| 1646 |
+
"img_path": "images/2c7ab5f672dbedeb5a93ad7409df68dfd0269aaaeb798e787db66c5317bf21db.jpg",
|
| 1647 |
+
"table_caption": [
|
| 1648 |
+
"Table 6: Background context for Shape. Shown is the Spearmann correlation coefficient (SROCC) of perceptual data vs. model-based MI prediction across shapes (i.e. considering all shapes rather than only Easy vs. Hard; note that the original robust finding the superiority of the Easy shape). Perceptual data from Weisstein & Harris (1974), where ”Day 1” and ”Days $2 { - } 4 ^ { \\dag }$ (averaged) are for the reduced-masking condition depicted in their Figure 3.) "
|
| 1649 |
+
],
|
| 1650 |
+
"table_footnote": [],
|
| 1651 |
+
"table_body": "<table><tr><td>Model</td><td></td><td>Day 1 Days 2-4</td><td>Masked</td></tr><tr><td>VGG-19</td><td>.36</td><td></td><td>.15</td></tr><tr><td>GoogLeNet</td><td>.31</td><td>.37 .22</td><td>.16</td></tr><tr><td>MRSA-152</td><td>.26</td><td>.26</td><td>.11</td></tr><tr><td>CaffeNet iter 1</td><td>.32</td><td></td><td>.39</td></tr><tr><td>CaffeNet iter 50K</td><td>.15</td><td>.29</td><td></td></tr><tr><td>CaffeNetiter310K</td><td>.16</td><td>.19</td><td>.16 .18</td></tr><tr><td></td><td>.26</td><td>.12</td><td>.48</td></tr><tr><td>Gabor Decomposition Steerable Pyramid</td><td>.24</td><td>.27 .32</td><td>.25</td></tr></table>",
|
| 1652 |
+
"bbox": [
|
| 1653 |
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308,
|
| 1654 |
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704,
|
| 1655 |
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687,
|
| 1656 |
+
848
|
| 1657 |
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],
|
| 1658 |
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"page_idx": 14
|
| 1659 |
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},
|
| 1660 |
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{
|
| 1661 |
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"type": "image",
|
| 1662 |
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"img_path": "images/8b57e7f7fd5822e264b389c7eed47b9194ccbdae2e14e5d0ad2c550462747c76.jpg",
|
| 1663 |
+
"image_caption": [
|
| 1664 |
+
"Figure 6: Images where predicted threshold is too high (”Overshoot”, where perturbation saliency is better than predicted) or too low (”Undershoot”), considered from several perceptual threshold ranges $\\pm 2$ dB of shown number). Some images are reproduced from Figure 1. "
|
| 1665 |
+
],
|
| 1666 |
+
"image_footnote": [],
|
| 1667 |
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"bbox": [
|
| 1668 |
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189,
|
| 1669 |
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174,
|
| 1670 |
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826,
|
| 1671 |
+
844
|
| 1672 |
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],
|
| 1673 |
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"page_idx": 15
|
| 1674 |
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},
|
| 1675 |
+
{
|
| 1676 |
+
"type": "image",
|
| 1677 |
+
"img_path": "images/6f9b1ef405c1200ecea4e9d17fea6192cb6189de656d2eabe5418e4163295a79.jpg",
|
| 1678 |
+
"image_caption": [
|
| 1679 |
+
"Figure 7: Background context for different DNN models (following figure 2). "
|
| 1680 |
+
],
|
| 1681 |
+
"image_footnote": [],
|
| 1682 |
+
"bbox": [
|
| 1683 |
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184,
|
| 1684 |
+
210,
|
| 1685 |
+
815,
|
| 1686 |
+
357
|
| 1687 |
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],
|
| 1688 |
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"page_idx": 16
|
| 1689 |
+
},
|
| 1690 |
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{
|
| 1691 |
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"type": "image",
|
| 1692 |
+
"img_path": "images/7138a34a90ab813f56ea5bb8f05be4794e2f461e3666d4db9eca006d09a47ca1.jpg",
|
| 1693 |
+
"image_caption": [
|
| 1694 |
+
"Figure 8: Background context for baseline DNN models (following figure 2). ”CaffeNet iter 310K” is reproduced from Figure 7. "
|
| 1695 |
+
],
|
| 1696 |
+
"image_footnote": [],
|
| 1697 |
+
"bbox": [
|
| 1698 |
+
174,
|
| 1699 |
+
569,
|
| 1700 |
+
694,
|
| 1701 |
+
810
|
| 1702 |
+
],
|
| 1703 |
+
"page_idx": 16
|
| 1704 |
+
},
|
| 1705 |
+
{
|
| 1706 |
+
"type": "image",
|
| 1707 |
+
"img_path": "images/27d167da9bfbd680de4450c36836d45b46cd94cb5fc4e9f608fdee72b4544a91.jpg",
|
| 1708 |
+
"image_caption": [
|
| 1709 |
+
"Figure 9: Background context for Shape. Shown for each model is the measured MI for the six ”Hard” shapes as a function of the MI for the ”Easy” shape. The last panel shows an analagous comparison measured in human subjects by Weisstein & Harris (1974). A data point which lies below the dashed diagonal indicates a configuration for which discriminating line location is easier for the Easy shape compared with the relevant Hard shape. "
|
| 1710 |
+
],
|
| 1711 |
+
"image_footnote": [],
|
| 1712 |
+
"bbox": [
|
| 1713 |
+
173,
|
| 1714 |
+
289,
|
| 1715 |
+
812,
|
| 1716 |
+
690
|
| 1717 |
+
],
|
| 1718 |
+
"page_idx": 17
|
| 1719 |
+
},
|
| 1720 |
+
{
|
| 1721 |
+
"type": "image",
|
| 1722 |
+
"img_path": "images/861dd0c87dd67be34fa74c6f7d5a517cb2e6e8d0d20c9adf7334aceb7112e790.jpg",
|
| 1723 |
+
"image_caption": [
|
| 1724 |
+
"Figure 10: Contrast sensitivity (following Figure 3) for DNN architectures CaffeNet, GoogLeNet, and ResNet-152. "
|
| 1725 |
+
],
|
| 1726 |
+
"image_footnote": [],
|
| 1727 |
+
"bbox": [
|
| 1728 |
+
173,
|
| 1729 |
+
272,
|
| 1730 |
+
825,
|
| 1731 |
+
743
|
| 1732 |
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],
|
| 1733 |
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"page_idx": 18
|
| 1734 |
+
},
|
| 1735 |
+
{
|
| 1736 |
+
"type": "image",
|
| 1737 |
+
"img_path": "images/74f90c0c817b0bea67bc6afa2e46b878b82a65de2628d7e46d194d25a2c10385.jpg",
|
| 1738 |
+
"image_caption": [
|
| 1739 |
+
"Figure 11: Comparison of contrast sensitivity. Shown are iso-output curves, for which perceived contrast is the same (Human), or for which the $L _ { 1 }$ change relative to a gray image is the same (DNN model VGG-19). To obtain a correspondence between human frequency values (given in cycles per degree of visual field) to DNN frequency values (given in cycles per image), a scaling was chosen such that the minima of the blue curve is given at the same frequency value. Human data is for subject M.A.G. as measured by Georgeson & Sullivan (1975). "
|
| 1740 |
+
],
|
| 1741 |
+
"image_footnote": [],
|
| 1742 |
+
"bbox": [
|
| 1743 |
+
171,
|
| 1744 |
+
329,
|
| 1745 |
+
808,
|
| 1746 |
+
622
|
| 1747 |
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],
|
| 1748 |
+
"page_idx": 19
|
| 1749 |
+
},
|
| 1750 |
+
{
|
| 1751 |
+
"type": "text",
|
| 1752 |
+
"text": "8 APPENDIX: EXPERIMENTAL SETUP ",
|
| 1753 |
+
"text_level": 1,
|
| 1754 |
+
"bbox": [
|
| 1755 |
+
174,
|
| 1756 |
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|
| 1757 |
+
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|
| 1758 |
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141
|
| 1759 |
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],
|
| 1760 |
+
"page_idx": 20
|
| 1761 |
+
},
|
| 1762 |
+
{
|
| 1763 |
+
"type": "text",
|
| 1764 |
+
"text": "8.1 DNN MODELS ",
|
| 1765 |
+
"text_level": 1,
|
| 1766 |
+
"bbox": [
|
| 1767 |
+
174,
|
| 1768 |
+
156,
|
| 1769 |
+
318,
|
| 1770 |
+
170
|
| 1771 |
+
],
|
| 1772 |
+
"page_idx": 20
|
| 1773 |
+
},
|
| 1774 |
+
{
|
| 1775 |
+
"type": "text",
|
| 1776 |
+
"text": "To collect DNN computation snapshots, we used MATLAB with MatConvNet version 1.0-beta20 (Vedaldi & Lenc, 2015). All MATLAB code will be made available upon acceptance of this manuscript. The pre-trained DNN models we have used are: CaffeNet (which is a variant of AlexNet provided in Caffe, Jia et al., 2014), GoogLeNet (Szegedy et al., 2014), VGG-19 (Simonyan & Zisserman, 2014), and ResNet-152 (He et al., 2015). The models were trained on the same ImageNet LSVRC. The CaffeNet model was trained using Caffe with the default ImageNet training parameters (stopping at iteration 310, 000) and imported into MatConvNet. For the GoogLeNet model, we used the imported pre-trained reference-Caffe implementation. For VGG-19 and ResNet-152, we used the imported pre-trained original versions. In all experiments input image size was $2 2 4 \\times 2 2 4$ or $2 2 7 \\times 2 2 7$ . ",
|
| 1777 |
+
"bbox": [
|
| 1778 |
+
173,
|
| 1779 |
+
183,
|
| 1780 |
+
825,
|
| 1781 |
+
320
|
| 1782 |
+
],
|
| 1783 |
+
"page_idx": 20
|
| 1784 |
+
},
|
| 1785 |
+
{
|
| 1786 |
+
"type": "text",
|
| 1787 |
+
"text": "8.2 BASELINE MODELS",
|
| 1788 |
+
"text_level": 1,
|
| 1789 |
+
"bbox": [
|
| 1790 |
+
176,
|
| 1791 |
+
338,
|
| 1792 |
+
348,
|
| 1793 |
+
352
|
| 1794 |
+
],
|
| 1795 |
+
"page_idx": 20
|
| 1796 |
+
},
|
| 1797 |
+
{
|
| 1798 |
+
"type": "text",
|
| 1799 |
+
"text": "As baselines to compare with pre-trained DNN models, we consider: (a) a multiscale linear filter bank of Gabor functions, (b) a steerable-pyramid linear filter bank (Simoncelli & Freeman, 1995), (c) the VGG-19 model for which the learned parameters (weights) were randomly scrambled within layer, and (d) the CaffeNet model at multiple time points during training. For the Gabor decomposition, the following Gabor filters were used: all compositions of $\\sigma = \\bar { \\{ 1 , 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \\} } \\mathrm { p x }$ , $\\lambda = \\{ 1 , 2 \\} \\cdot \\sigma$ , orientation $\\underline { { \\underline { { \\mathbf { \\Pi } } } } } = \\{ 0 , \\pi / 3 , 2 \\pi / 3 , \\pi , 4 \\pi / 3 , 5 \\pi / 3 \\}$ , and phase $= \\{ 0 , \\pi / 2 \\}$ . ",
|
| 1800 |
+
"bbox": [
|
| 1801 |
+
174,
|
| 1802 |
+
364,
|
| 1803 |
+
825,
|
| 1804 |
+
449
|
| 1805 |
+
],
|
| 1806 |
+
"page_idx": 20
|
| 1807 |
+
},
|
| 1808 |
+
{
|
| 1809 |
+
"type": "text",
|
| 1810 |
+
"text": "8.3 IMAGE PERTURBATION EXPERIMENT ",
|
| 1811 |
+
"text_level": 1,
|
| 1812 |
+
"bbox": [
|
| 1813 |
+
176,
|
| 1814 |
+
465,
|
| 1815 |
+
465,
|
| 1816 |
+
479
|
| 1817 |
+
],
|
| 1818 |
+
"page_idx": 20
|
| 1819 |
+
},
|
| 1820 |
+
{
|
| 1821 |
+
"type": "text",
|
| 1822 |
+
"text": "The noiseless images were obtained from Alam et al. (2014). In main text, ”image scale” refers to percent coverage of DNN input. Since size of original images $( 1 4 9 \\times 1 4 9 )$ is smaller than DNN input of $( 2 2 4 \\times 2 2 4 )$ or $( 2 2 7 \\times 2 2 7 )$ ), the images were resized by a factor of 1.5 so that $100 \\%$ image scale covers approximately the entire DNN input area. ",
|
| 1823 |
+
"bbox": [
|
| 1824 |
+
174,
|
| 1825 |
+
492,
|
| 1826 |
+
825,
|
| 1827 |
+
547
|
| 1828 |
+
],
|
| 1829 |
+
"page_idx": 20
|
| 1830 |
+
},
|
| 1831 |
+
{
|
| 1832 |
+
"type": "text",
|
| 1833 |
+
"text": "Human psychophysics and DNN experiments were done for nearly identical images. A slight discrepancy relates to how the image is blended with the background in the special case where the region where noise is added has no image surround at one or two side. In these sides (which depend on the technical procedure with which images were obtained, see Alam et al., 2014), the surround blending here was hard, while the original was smooth. ",
|
| 1834 |
+
"bbox": [
|
| 1835 |
+
174,
|
| 1836 |
+
555,
|
| 1837 |
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|
| 1838 |
+
625
|
| 1839 |
+
],
|
| 1840 |
+
"page_idx": 20
|
| 1841 |
+
},
|
| 1842 |
+
{
|
| 1843 |
+
"type": "text",
|
| 1844 |
+
"text": "8.4 BACKGROUND CONTEXT EXPERIMENT ",
|
| 1845 |
+
"text_level": 1,
|
| 1846 |
+
"bbox": [
|
| 1847 |
+
176,
|
| 1848 |
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642,
|
| 1849 |
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480,
|
| 1850 |
+
655
|
| 1851 |
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],
|
| 1852 |
+
"page_idx": 20
|
| 1853 |
+
},
|
| 1854 |
+
{
|
| 1855 |
+
"type": "text",
|
| 1856 |
+
"text": "8.4.1 SEGMENTATION",
|
| 1857 |
+
"text_level": 1,
|
| 1858 |
+
"bbox": [
|
| 1859 |
+
174,
|
| 1860 |
+
667,
|
| 1861 |
+
339,
|
| 1862 |
+
683
|
| 1863 |
+
],
|
| 1864 |
+
"page_idx": 20
|
| 1865 |
+
},
|
| 1866 |
+
{
|
| 1867 |
+
"type": "text",
|
| 1868 |
+
"text": "The images used are based on the Texture Discrimination Task (Karni & Sagi, 1991). In the variant considered here (Pinchuk-Yacobi et al., 2015), subjects were presented with a grid of lines, all of which were horizontal, except two or three that were diagonal. Subjects discriminated whether the arrangement of diagonal lines is horizontal or vertical, and this discrimination was found to be more difficult when the central line is horizontal rather than diagonal (”Hard” vs. ”Easy” in Figure 2a). To limit human performance in this task, two manipulations were applied: (a) the location of each line in the pattern was jittered, and (b) a noise mask was presented briefly after the pattern. Here we only retained (a). ",
|
| 1869 |
+
"bbox": [
|
| 1870 |
+
174,
|
| 1871 |
+
691,
|
| 1872 |
+
825,
|
| 1873 |
+
804
|
| 1874 |
+
],
|
| 1875 |
+
"page_idx": 20
|
| 1876 |
+
},
|
| 1877 |
+
{
|
| 1878 |
+
"type": "text",
|
| 1879 |
+
"text": "A total of 90 configurations were tested, obtained by combinations of the following alternatives: ",
|
| 1880 |
+
"bbox": [
|
| 1881 |
+
174,
|
| 1882 |
+
810,
|
| 1883 |
+
802,
|
| 1884 |
+
825
|
| 1885 |
+
],
|
| 1886 |
+
"page_idx": 20
|
| 1887 |
+
},
|
| 1888 |
+
{
|
| 1889 |
+
"type": "text",
|
| 1890 |
+
"text": "• Three scales: line length of 9, 12.3, or $1 9 . 4 \\ \\mathrm { p x }$ (number of lines co-varied with line length, see Figure 12). \n• Three levels of location jittering, defined as a multiple of line length: $\\{ 1 , 2 , 3 \\} \\cdot 0 . 0 6 2 5 \\cdot l$ px, where $l$ is the length of a line in the pattern. Jittering was applied separately to each line in the pattern. \n• Ten locations of diagonal lines: center, random, four locations of half-distance from center to corners, four locations of half-distance from center to image borders. ",
|
| 1891 |
+
"bbox": [
|
| 1892 |
+
217,
|
| 1893 |
+
838,
|
| 1894 |
+
825,
|
| 1895 |
+
946
|
| 1896 |
+
],
|
| 1897 |
+
"page_idx": 20
|
| 1898 |
+
},
|
| 1899 |
+
{
|
| 1900 |
+
"type": "text",
|
| 1901 |
+
"text": "For each configuration, the discriminated arrangement of diagonal lines was either horizontal or vertical, and the central line was either horizontal or diagonal (i.e. hard or easy). ",
|
| 1902 |
+
"bbox": [
|
| 1903 |
+
173,
|
| 1904 |
+
126,
|
| 1905 |
+
825,
|
| 1906 |
+
155
|
| 1907 |
+
],
|
| 1908 |
+
"page_idx": 21
|
| 1909 |
+
},
|
| 1910 |
+
{
|
| 1911 |
+
"type": "image",
|
| 1912 |
+
"img_path": "images/e61f99d55dac1fec445d942c16502244e20273274c1c7ec161dba0ab42e4345e.jpg",
|
| 1913 |
+
"image_caption": [
|
| 1914 |
+
"Figure 12: Pattern scales used in the different configurations of the Segmentation condition. Actual images used were white-on-black rather than black-on-white. "
|
| 1915 |
+
],
|
| 1916 |
+
"image_footnote": [],
|
| 1917 |
+
"bbox": [
|
| 1918 |
+
253,
|
| 1919 |
+
171,
|
| 1920 |
+
745,
|
| 1921 |
+
295
|
| 1922 |
+
],
|
| 1923 |
+
"page_idx": 21
|
| 1924 |
+
},
|
| 1925 |
+
{
|
| 1926 |
+
"type": "text",
|
| 1927 |
+
"text": "8.4.2 CROWDING",
|
| 1928 |
+
"text_level": 1,
|
| 1929 |
+
"bbox": [
|
| 1930 |
+
174,
|
| 1931 |
+
377,
|
| 1932 |
+
310,
|
| 1933 |
+
392
|
| 1934 |
+
],
|
| 1935 |
+
"page_idx": 21
|
| 1936 |
+
},
|
| 1937 |
+
{
|
| 1938 |
+
"type": "text",
|
| 1939 |
+
"text": "The images used are motivated by the crowding effect (Livne & Sagi, 2007; Pelli et al., 2004). ",
|
| 1940 |
+
"bbox": [
|
| 1941 |
+
176,
|
| 1942 |
+
401,
|
| 1943 |
+
790,
|
| 1944 |
+
417
|
| 1945 |
+
],
|
| 1946 |
+
"page_idx": 21
|
| 1947 |
+
},
|
| 1948 |
+
{
|
| 1949 |
+
"type": "text",
|
| 1950 |
+
"text": "A total of 90 configurations were tested, obtained by combinations of the following alternatives: ",
|
| 1951 |
+
"bbox": [
|
| 1952 |
+
178,
|
| 1953 |
+
422,
|
| 1954 |
+
802,
|
| 1955 |
+
439
|
| 1956 |
+
],
|
| 1957 |
+
"page_idx": 21
|
| 1958 |
+
},
|
| 1959 |
+
{
|
| 1960 |
+
"type": "text",
|
| 1961 |
+
"text": "• Three scales: font size of 15.1, 20.6, or $3 2 . 4 { \\mathrm { p x } }$ (see Figure 13). \n• Three levels of discriminated-letter location jittering, defined as a multiple of font size: $\\{ 1 , 2 , 3 \\} \\cdot 0 . 0 6 2 5 \\cdot l$ px, where $l$ is font size. The jitter of surround letters (M, N, S, and T) was fixed (i.e. the background was static). \n• Ten locations: center, random, four locations of half-distance from center to corners, four locations of half-distance from center to image borders. ",
|
| 1962 |
+
"bbox": [
|
| 1963 |
+
215,
|
| 1964 |
+
450,
|
| 1965 |
+
825,
|
| 1966 |
+
546
|
| 1967 |
+
],
|
| 1968 |
+
"page_idx": 21
|
| 1969 |
+
},
|
| 1970 |
+
{
|
| 1971 |
+
"type": "text",
|
| 1972 |
+
"text": "For each configuration, the discriminated letter was either A, B, C, D, E, or F, and the background was either blank (easy) or composed of the letters M, N, S, and T (hard). ",
|
| 1973 |
+
"bbox": [
|
| 1974 |
+
174,
|
| 1975 |
+
560,
|
| 1976 |
+
826,
|
| 1977 |
+
588
|
| 1978 |
+
],
|
| 1979 |
+
"page_idx": 21
|
| 1980 |
+
},
|
| 1981 |
+
{
|
| 1982 |
+
"type": "image",
|
| 1983 |
+
"img_path": "images/3b00649747048479281c6d49a31d32f1e301675100b165b5e0554a56148e123c.jpg",
|
| 1984 |
+
"image_caption": [
|
| 1985 |
+
"Figure 13: Pattern scales used in the different configurations of the Crowding condition. Actual images used were white-on-black rather than black-on-white. "
|
| 1986 |
+
],
|
| 1987 |
+
"image_footnote": [],
|
| 1988 |
+
"bbox": [
|
| 1989 |
+
253,
|
| 1990 |
+
604,
|
| 1991 |
+
745,
|
| 1992 |
+
729
|
| 1993 |
+
],
|
| 1994 |
+
"page_idx": 21
|
| 1995 |
+
},
|
| 1996 |
+
{
|
| 1997 |
+
"type": "text",
|
| 1998 |
+
"text": "8.4.3 SHAPE ",
|
| 1999 |
+
"text_level": 1,
|
| 2000 |
+
"bbox": [
|
| 2001 |
+
174,
|
| 2002 |
+
810,
|
| 2003 |
+
276,
|
| 2004 |
+
825
|
| 2005 |
+
],
|
| 2006 |
+
"page_idx": 21
|
| 2007 |
+
},
|
| 2008 |
+
{
|
| 2009 |
+
"type": "text",
|
| 2010 |
+
"text": "The images used are based on the object superiority effect by Weisstein & Harris (1974), where discriminating a line location is easier when combined with surrounding lines a shape is formed. ",
|
| 2011 |
+
"bbox": [
|
| 2012 |
+
176,
|
| 2013 |
+
835,
|
| 2014 |
+
823,
|
| 2015 |
+
864
|
| 2016 |
+
],
|
| 2017 |
+
"page_idx": 21
|
| 2018 |
+
},
|
| 2019 |
+
{
|
| 2020 |
+
"type": "text",
|
| 2021 |
+
"text": "A total of 90 configurations were tested, obtained by combinations of the following alternatives: ",
|
| 2022 |
+
"bbox": [
|
| 2023 |
+
174,
|
| 2024 |
+
871,
|
| 2025 |
+
802,
|
| 2026 |
+
886
|
| 2027 |
+
],
|
| 2028 |
+
"page_idx": 21
|
| 2029 |
+
},
|
| 2030 |
+
{
|
| 2031 |
+
"type": "text",
|
| 2032 |
+
"text": "• Three scales: discriminated-line length of 9, 15.1, or 22.7 px (see Figure 14). • Five levels of whole-pattern location jittering, defined as a multiple of discriminated-line length: $\\{ 1 , 2 , 5 , 1 0 , 1 \\bar { 5 } \\} \\cdot 0 . 0 6 2 5 \\cdot l$ px, where $l$ is the length of the discriminated line. ",
|
| 2033 |
+
"bbox": [
|
| 2034 |
+
217,
|
| 2035 |
+
897,
|
| 2036 |
+
825,
|
| 2037 |
+
946
|
| 2038 |
+
],
|
| 2039 |
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"page_idx": 21
|
| 2040 |
+
},
|
| 2041 |
+
{
|
| 2042 |
+
"type": "text",
|
| 2043 |
+
"text": "• Six ”hard” background line layouts (patterns $b { - } f$ of their Figure 2 and the additional pattern $f$ of their Figure 3 in Weisstein & Harris, 1974). The ”easy” layout was always the same (pattern $a$ ). ",
|
| 2044 |
+
"bbox": [
|
| 2045 |
+
215,
|
| 2046 |
+
126,
|
| 2047 |
+
825,
|
| 2048 |
+
167
|
| 2049 |
+
],
|
| 2050 |
+
"page_idx": 22
|
| 2051 |
+
},
|
| 2052 |
+
{
|
| 2053 |
+
"type": "text",
|
| 2054 |
+
"text": "For each configuration, the line whose location is discriminated had four possible locations (two locations are shown in Figure 2c), and the surrounding background line layout could compose a shape (easy) or not (hard). ",
|
| 2055 |
+
"bbox": [
|
| 2056 |
+
176,
|
| 2057 |
+
181,
|
| 2058 |
+
825,
|
| 2059 |
+
224
|
| 2060 |
+
],
|
| 2061 |
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"page_idx": 22
|
| 2062 |
+
},
|
| 2063 |
+
{
|
| 2064 |
+
"type": "image",
|
| 2065 |
+
"img_path": "images/3a4cd4d746343780ed7fd9d056ac0f854d4db1895ecffe90c3de68ac7286044d.jpg",
|
| 2066 |
+
"image_caption": [],
|
| 2067 |
+
"image_footnote": [],
|
| 2068 |
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"bbox": [
|
| 2069 |
+
254,
|
| 2070 |
+
241,
|
| 2071 |
+
745,
|
| 2072 |
+
364
|
| 2073 |
+
],
|
| 2074 |
+
"page_idx": 22
|
| 2075 |
+
},
|
| 2076 |
+
{
|
| 2077 |
+
"type": "text",
|
| 2078 |
+
"text": "Figure 14: Pattern scales used in the different configurations of the Shape condition. Actual images used were white-on-black rather than black-on-white. ",
|
| 2079 |
+
"bbox": [
|
| 2080 |
+
173,
|
| 2081 |
+
393,
|
| 2082 |
+
823,
|
| 2083 |
+
422
|
| 2084 |
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],
|
| 2085 |
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"page_idx": 22
|
| 2086 |
+
},
|
| 2087 |
+
{
|
| 2088 |
+
"type": "text",
|
| 2089 |
+
"text": "8.5 CONTRAST SENSITIVITY EXPERIMENT ",
|
| 2090 |
+
"text_level": 1,
|
| 2091 |
+
"bbox": [
|
| 2092 |
+
176,
|
| 2093 |
+
445,
|
| 2094 |
+
478,
|
| 2095 |
+
458
|
| 2096 |
+
],
|
| 2097 |
+
"page_idx": 22
|
| 2098 |
+
},
|
| 2099 |
+
{
|
| 2100 |
+
"type": "text",
|
| 2101 |
+
"text": "Used images depicted sine gratings at different contrast, spatial frequency, sine phase, and sine orientation combinations. ",
|
| 2102 |
+
"bbox": [
|
| 2103 |
+
174,
|
| 2104 |
+
469,
|
| 2105 |
+
825,
|
| 2106 |
+
498
|
| 2107 |
+
],
|
| 2108 |
+
"page_idx": 22
|
| 2109 |
+
}
|
| 2110 |
+
]
|
parse/train/BJbD_Pqlg/BJbD_Pqlg_middle.json
ADDED
|
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See raw diff
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|
|
parse/train/BJbD_Pqlg/BJbD_Pqlg_model.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
parse/train/BJeWUs05KQ/BJeWUs05KQ_layout.pdf
ADDED
|
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|
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size 2713081
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parse/train/BJeWUs05KQ/BJeWUs05KQ_origin.pdf
ADDED
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version https://git-lfs.github.com/spec/v1
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|
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size 2544012
|
parse/train/BJeWUs05KQ/BJeWUs05KQ_span.pdf
ADDED
|
@@ -0,0 +1,3 @@
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| 1 |
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version https://git-lfs.github.com/spec/v1
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|
| 3 |
+
size 2717637
|
parse/train/BJj6qGbRW/BJj6qGbRW.md
ADDED
|
@@ -0,0 +1,346 @@
|
|
|
|
|
|
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|
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|
|
|
|
| 1 |
+
# FEW-SHOT LEARNING WITH GRAPH NEURAL NETWORKS
|
| 2 |
+
|
| 3 |
+
Victor Garcia∗ Amsterdam Machine Learning Lab University of Amsterdam Amsterdam, 1098 XH, NL v.garciasatorras@uva.nl
|
| 4 |
+
|
| 5 |
+
Joan Bruna
|
| 6 |
+
Courant Institute of Mathematical Sciences
|
| 7 |
+
New York University
|
| 8 |
+
New York City, NY, 10010, USA
|
| 9 |
+
bruna@cims.nyu.edu
|
| 10 |
+
|
| 11 |
+
# ABSTRACT
|
| 12 |
+
|
| 13 |
+
We propose to study the problem of few-shot learning with the prism of inference on a partially observed graphical model, constructed from a collection of input images whose label can be either observed or not. By assimilating generic message-passing inference algorithms with their neural-network counterparts, we define a graph neural network architecture that generalizes several of the recently proposed few-shot learning models. Besides providing improved numerical performance, our framework is easily extended to variants of few-shot learning, such as semi-supervised or active learning, demonstrating the ability of graph-based models to operate well on ‘relational’ tasks.
|
| 14 |
+
|
| 15 |
+
# 1 INTRODUCTION
|
| 16 |
+
|
| 17 |
+
Supervised end-to-end learning has been extremely successful in computer vision, speech, or machine translation tasks, thanks to improvements in optimization technology, larger datasets and streamlined designs of deep convolutional or recurrent architectures. Despite these successes, this learning setup does not cover many aspects where learning is nonetheless possible and desirable.
|
| 18 |
+
|
| 19 |
+
One such instance is the ability to learn from few examples, in the so-called few-shot learning tasks. Rather than relying on regularization to compensate for the lack of data, researchers have explored ways to leverage a distribution of similar tasks, inspired by human learning Lake et al. (2015). This defines a new supervised learning setup (also called ‘meta-learning’) in which the input-output pairs are no longer given by iid samples of images and their associated labels, but by iid samples of collections of images and their associated label similarity.
|
| 20 |
+
|
| 21 |
+
A recent and highly-successful research program has exploited this meta-learning paradigm on the few-shot image classification task Lake et al. (2015); Koch et al. (2015); Vinyals et al. (2016); Mishra et al. (2017); Snell et al. (2017). In essence, these works learn a contextual, task-specific similarity measure, that first embeds input images using a CNN, and then learns how to combine the embedded images in the collection to propagate the label information towards the target image.
|
| 22 |
+
|
| 23 |
+
In particular, Vinyals et al. (2016) cast the few-shot learning problem as a supervised classification task mapping a support set of images into the desired label, and developed an end-to-end architecture accepting those support sets as input via attention mechanisms. In this work, we build upon this line of work, and argue that this task is naturally expressed as a supervised interpolation problem on a graph, where nodes are associated with the images in the collection, and edges are given by a trainable similarity kernels. Leveraging recent progress on representation learning for graphstructured data Bronstein et al. (2017); Gilmer et al. (2017), we thus propose a simple graph-based few-shot learning model that implements a task-driven message passing algorithm. The resulting architecture is trained end-to-end, captures the invariances of the task, such as permutations within the input collections, and offers a good tradeoff between simplicity, generality, performance and sample complexity.
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+
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Besides few-shot learning, a related task is the ability to learn from a mixture of labeled and unlabeled examples — semi-supervised learning, as well as active learning, in which the learner has the option to request those missing labels that will be most helpful for the prediction task. Our graphbased architecture is naturally extended to these setups with minimal changes in the training design. We validate experimentally the model on few-shot image classification, matching state-of-the-art performance with considerably fewer parameters, and demonstrate applications to semi-supervised and active learning setups.
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Our contributions are summarized as follows:
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• We cast few-shot learning as a supervised message passing task which is trained end-to-end using graph neural networks.
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• We match state-of-the-art performance on Omniglot and Mini-Imagenet tasks with fewer parameters.
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• We extend the model in the semi-supervised and active learning regimes.
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+
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The rest of the paper is structured as follows. Section 2 describes related work, Sections 3, 4 and 5 present the problem setup, our graph neural network model and the training, and Section 6 reports numerical experiments.
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# 2 RELATED WORK
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One-shot learning was first introduced by Fei-Fei et al. (2006), they assumed that currently learned classes can help to make predictions on new ones when just one or few labels are available. More recently, Lake et al. (2015) presented a Hierarchical Bayesian model that reached human level error on few-shot learning alphabet recongition tasks.
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Since then, great progress has been done in one-shot learning. Koch et al. (2015) presented a deeplearning model based on computing the pair-wise distance between samples using Siamese Networks, then, this learned distance can be used to solve one-shot problems by $\mathbf { k }$ -nearest neighbors classification. Vinyals et al. (2016) Presented an end-to-end trainable $\mathbf { k }$ -nearest neighbors using the cosine distance, they also introduced a contextual mechanism using an attention LSTM model that takes into account all the samples of the subset $\tau$ when computing the pair-wise distance between samples. Snell et al. (2017) extended the work from Vinyals et al. (2016), by using euclidean distance instead of cosine which provided significant improvements, they also build a prototype representation of each class for the few-shot learning scenario. Mehrotra & Dukkipati (2017) trained a deep residual network together with a generative model to approximate the pair-wise distance between samples.
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A new line of meta-learners for one-shot learning is rising lately: Ravi & Larochelle (2016) introduced a meta-learning method where an LSTM updates the weights of a classifier for a given episode. Munkhdalai & Yu (2017) also presented a meta-learning architecture that learns meta-level knowledge across tasks, and it changes its inductive bias via fast parametrization. Finn et al. (2017) is using a model agnostic meta-learner based on gradient descent, the goal is to train a classification model such that given a new task, a small amount of gradient steps with few data will be enough to generalize. Lately, Mishra et al. (2017) used Temporal Convolutions which are deep recurrent networks based on dilated convolutions, this method also exploits contextual information from the subset $\tau$ providing very good results.
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+
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Another related area of research concerns deep learning architectures on graph-structured data. The GNN was first proposed in Gori et al. (2005); Scarselli et al. (2009), as a trainable recurrent messagepassing whose fixed points could be adjusted discriminatively. Subsequent works Li et al. (2015); Sukhbaatar et al. (2016) have relaxed the model by untying the recurrent layer weights and proposed several nonlinear updates through gating mechanisms. Graph neural networks are in fact natural generalizations of convolutional networks to non-Euclidean graphs. Bruna et al. (2013); Henaff et al. (2015) proposed to learn smooth spectral multipliers of the graph Laplacian, albeit with high computational cost, and Defferrard et al. (2016); Kipf & Welling (2016) resolved the computational bottleneck by learning polynomials of the graph Laplacian, thus avoiding the computation of eigenvectors and completing the connection with GNNs. In particular, Kipf & Welling (2016) was the first to propose the use of GNNs on semi-supervised classification problems. We refer the reader to Bronstein et al. (2017) for an exhaustive literature review on the topic. GNNs and the analogous Neural Message Passing Models are finding application in many different domains. Battaglia et al.
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(2016); Chang et al. (2016) develop graph interaction networks that learn pairwise particle interactions and apply them to discrete particle physical dynamics. Duvenaud et al. (2015); Kearnes et al. (2016) study molecular fingerprints using variants of the GNN architecture, and Gilmer et al. (2017) further develop the model by combining it with set representations Vinyals et al. (2015), showing state-of-the-art results on molecular prediction.
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# 3 PROBLEM SET-UP
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We describe first the general setup and notations, and then particularize it to the case of few-shot learning, semi-supervised learning and active learning.
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We consider input-output pairs $( \mathcal { T } _ { i } , Y _ { i } ) _ { i }$ drawn iid from a distribution $P$ of partially-labeled image collections
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+
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+
$$
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\begin{array} { r c l } { { { \cal T } } } & { { = } } & { { \{ \{ ( x _ { 1 } , l _ { 1 } ) , \dots ( x _ { s } , l _ { s } ) \} , \{ \tilde { x } _ { 1 } , \dots , \tilde { x } _ { r } \} , \{ \bar { x } _ { 1 } , \dots , \bar { x } _ { t } \} ; l _ { i } \in \{ 1 , K \} , x _ { i } , \tilde { x } _ { j } , \bar { x } _ { j } \sim \{ \mathcal { P } _ { l } ( \mathbb { R } ^ { N } ) \} \} , } } \\ { { { \cal Y } } } & { { = } } & { { ( y _ { 1 } , \dots , y _ { t } ) \in \{ 1 , K \} ^ { t } , } } \\ { { } } & { { } } & { { ( 1 ) } } \end{array}
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+
$$
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for arbitrary values of $s , r , t$ and $K$ . Where $s$ is the number of labeled samples, $r$ is the number of unlabeled samples $r > 0$ for the semi-supervised and active learning scenarios) and $t$ is the number of samples to classify. $K$ is the number of classes. We will focus in the case $t = 1$ where we just classify one sample per task $\tau$ . $\mathcal { P } _ { l } ( \mathbb { R } ^ { N } )$ denotes a class-specific image distribution over $\mathbb { R } ^ { N }$ . In our context, the targets $Y _ { i }$ are associated with image categories of designated images $\bar { x } _ { 1 } , \ldots , \bar { x } _ { t } \in \mathcal { T } _ { i }$ with no observed label. Given a training set $\{ ( \mathcal { T } _ { i } , Y _ { i } ) _ { i } \} _ { i \leq L }$ , we consider the standard supervised learning objective
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+
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$$
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\operatorname* { m i n } _ { \Theta } \frac { 1 } { L } \sum _ { i \leq L } \ell ( \Phi ( \mathcal { T } _ { i } ; \Theta ) , Y _ { i } ) + \mathcal { R } ( \Theta ) ,
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+
$$
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+
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using the model $\Phi ( { \mathcal { T } } ; \Theta ) = p ( Y \mid { \mathcal { T } } )$ specified in Section 4 and $\mathcal { R }$ is a standard regularization objective.
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+
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Few-Shot Learning When $r = 0$ , $t = 1$ and $s = q K$ , there is a single image in the collection with unknown label. If moreover each label appears exactly $q$ times, this setting is referred as the $q$ -shot, $K$ -way learning.
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+
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Semi-Supervised Learning When $r > 0$ and $t = 1$ , the input collection contains auxiliary images $\tilde { x } _ { 1 } , \ldots , \tilde { x } _ { r }$ that the model can use to improve the prediction accuracy, by leveraging the fact that these samples are drawn from common distributions as those determining the output.
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+
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Active Learning In the active learning setting, the learner has the ability to request labels from the sub-collection $\{ \tilde { x } _ { 1 } , \ldots , \tilde { x } _ { r } \}$ . We are interested in studying to what extent this active learning can improve the performance with respect to the previous semi-supervised setup, and match the performance of the one-shot learning setting with $s _ { 0 }$ known labels when $s + r = s _ { 0 }$ , $s \ll s _ { 0 }$ .
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# 4 MODEL
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This section presents our approach, based on a simple end-to-end graph neural network architecture. We first explain how the input context is mapped into a graphical representation, then detail the architecture, and next show how this model generalizes a number of previously published few-shot learning architectures.
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# 4.1 SET AND GRAPH INPUT REPRESENTATIONS
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+
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The input $\tau$ contains a collection of images, both labeled and unlabeled. The goal of few-shot learning is to propagate label information from labeled samples towards the unlabeled query image. This propagation of information can be formalized as a posterior inference over a graphical model determined by the input images and labels.
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+
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Following several recent works that cast posterior inference using message passing with neural networks defined over graphs Scarselli et al. (2009); Duvenaud et al. (2015); Gilmer et al. (2017), we associate $\tau$ with a fully-connected graph $G _ { \mathcal { T } } = ( V , E )$ where nodes $v _ { a } \in V$ correspond to the images present in $\tau$ (both labeled and unlabeled). In this context, the setup does not specify a fixed similarity $e _ { a , a ^ { \prime } }$ between images $x _ { a }$ and $x _ { a ^ { \prime } }$ , suggesting an approach where this similarity measure is learnt in a discriminative fashion with a parametric model similarly as in Gilmer et al. (2017), such as a siamese neural architecture. This framework is closely related to the set representation from Vinyals et al. (2016), but extends the inference mechanism using the graph neural network formalism that we detail next.
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+
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+
Figure 1: Visual representation of One-Shot Learning setting.
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+
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+
# 4.2 GRAPH NEURAL NETWORKS
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Graph Neural Networks, introduced in Gori et al. (2005); Scarselli et al. (2009) and further simplified in Li et al. (2015); Duvenaud et al. (2015); Sukhbaatar et al. (2016) are neural networks based on local operators of a graph $G = ( V , E )$ , offering a powerful balance between expressivity and sample complexity; see Bronstein et al. (2017) for a recent survey on models and applications of deep learning on graphs.
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In its simplest incarnation, given an input signal $F \in \mathbb { R } ^ { V \times d }$ on the vertices of a weighted graph $G$ , we consider a family $\mathcal { A }$ of graph intrinsic linear operators that act locally on this signal. The simplest is the adjacency operator $\mathsf { \bar { A } } : \mathsf { F } \mapsto A ( F )$ where $\begin{array} { r } { ( A F ) _ { i } : = \sum _ { j \sim i } w _ { i , j } \mathbf { \bar { F } } _ { j } } \end{array}$ , with $i \sim j$ iff $( i , j ) \in E$ and $w _ { i , j }$ its associated weight. A GNN layer $\operatorname { G c } ( \cdot )$ receives as input a signal $\mathbf { x } ^ { ( k ) } \in \mathbb { R } ^ { V \times d _ { k } }$ and produces $\mathbf { x } ^ { ( k + 1 ) } \in \mathbb { R } ^ { V \times d _ { k + 1 } }$ as
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+
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+
$$
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+
{ \bf x } _ { l } ^ { ( k + 1 ) } = { \bf G } { \bf c } ( { \bf x } ^ { ( k ) } ) = \rho \left( \sum _ { B \in \mathcal { A } } B { \bf x } ^ { ( k ) } \theta _ { B , l } ^ { ( k ) } \right) \mathrm { ~ , ~ } l = d _ { 1 } \mathrm { ~ . ~ . ~ } d _ { k + 1 } \mathrm { ~ , ~ }
|
| 92 |
+
$$
|
| 93 |
+
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+
where $\Theta = \{ \theta _ { 1 } ^ { ( k ) } , \dots , \theta _ { | \mathcal { A } | } ^ { ( k ) } \} _ { k }$ , ${ \boldsymbol { \theta } _ { B } ^ { ( k ) } \in \mathbb { R } ^ { d _ { k } \times d _ { k + 1 } } }$ , are trainable parameters and $\rho ( \cdot )$ is a point-wise non-linearity, chosen in this work to be a ‘leaky’ ReLU et al. (2015).
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+
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+
Authors have explored several modeling variants from this basic formulation, by replacing the pointwise nonlinearity with gating operations Duvenaud et al. (2015), or by generalizing the generator family to Laplacian polynomials Defferrard et al. (2016); Kipf & Welling (2016); Bruna et al. (2013), or including $2 ^ { J }$ -th powers of $A$ to $\mathcal { A }$ , $A _ { J } = \operatorname* { m i n } ( 1 , A ^ { 2 ^ { J } } )$ to encode $2 ^ { J }$ -hop neighborhoods of each node Bruna & Li (2017). Cascaded operations in the form (2) are able to approximate a wide range of graph inference tasks. In particular, inspired by message-passing algorithms, Kearnes et al. (2016); Gilmer et al. (2017) generalized the GNN to also learn edge features $\tilde { A } ^ { ( k ) }$ from the current node hidden representation:
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+
|
| 98 |
+
$$
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+
\tilde { A } _ { i , j } ^ { ( k ) } = \varphi _ { \tilde { \theta } } ( \mathbf { x } _ { i } ^ { ( k ) } , \mathbf { x } _ { j } ^ { ( k ) } ) ,
|
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+
$$
|
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+
|
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+
|
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+
Figure 2: Graph Neural Network ilustration. The Adjacency matrix is computed before every Convolutional Layer.
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+
|
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+
where $\varphi$ is a symmetric function parametrized with e.g. a neural network. In this work, we consider a Multilayer Perceptron stacked after the absolute difference between two vector nodes. See eq. 4:
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+
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+
$$
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+
\varphi _ { \tilde { \theta } } ( \mathbf { x } _ { i } ^ { ( k ) } , \mathbf { x } _ { j } ^ { ( k ) } ) = \mathbf { M L P } _ { \tilde { \theta } } ( a b s ( \mathbf { x } _ { i } ^ { ( k ) } - \mathbf { x } _ { j } ^ { ( k ) } ) )
|
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+
$$
|
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+
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+
Then $\varphi$ is a metric, which is learned by doing a non-linear combination of the absolute difference between the individual features of two nodes. Using this architecture the distance property Symmetry $\varphi _ { \tilde { \theta } } ( a , b ) = \varphi _ { \tilde { \theta } } ( b , a )$ is fulfilled by construction and the distance property Identity $\varphi _ { \tilde { \theta } } ( a , a ) = 0$ is easily learned.
|
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+
|
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+
The trainable adjacency is then normalized to a stochastic kernel by using a softmax along each row. The resulting update rules for node features are obtained by adding the edge feature kernel $\tilde { A } ^ { ( k ) }$ into the generator family $\mathcal { A } = \{ \tilde { A } ^ { ( k ) } , \mathbf { 1 } \}$ and applying (2). Adjacency learning is particularly important in applications where the input set is believed to have some geometric structure, but the metric is not known a priori, such as is our case.
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+
|
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+
In general graphs, the network depth is chosen to be of the order of the graph diameter, so that all nodes obtain information from the entire graph. In our context, however, since the graph is densely connected, the depth is interpreted simply as giving the model more expressive power.
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+
Construction of Initial Node Features The input collection $\tau$ is mapped into node features as follows. For images $x _ { i } \in \mathcal T$ with known label $l _ { i }$ , the one-hot encoding of the label is concatenated with the embedding features of the image at the input of the GNN.
|
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+
|
| 119 |
+
$$
|
| 120 |
+
\mathbf { x } _ { i } ^ { ( 0 ) } = \left( \phi ( { x } _ { i } ) , h ( l _ { i } ) \right) ,
|
| 121 |
+
$$
|
| 122 |
+
|
| 123 |
+
where $\phi$ is a Convolutional neural network and $h ( l ) \ \in \ \mathbb { R } _ { + } ^ { K }$ is a one-hot encoding of the label. Architectural details for $\phi$ are detailed in Section 6.1.1 and 6.1.2. For images $\tilde { x } _ { j } , \bar { x } _ { j ^ { \prime } }$ with unknown label $l _ { i }$ , we modify the previous construction to account for full uncertainty about the label variable by replacing $h ( l )$ with the uniform distribution over the $K$ -simplex: $V _ { j } \doteq ( \phi ( \tilde { x } _ { j } ) , K ^ { - 1 } \mathbf { 1 } _ { K } )$ , and analogously for $\bar { x }$ .
|
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+
|
| 125 |
+
# 4.3 RELATIONSHIP WITH EXISTING MODELS
|
| 126 |
+
|
| 127 |
+
The graph neural network formulation of few-shot learning generalizes a number of recent models proposed in the literature.
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+
Siamese Networks Siamese Networks Koch et al. (2015) can be interpreted as a single layer message-passing iteration of our model, and using the same initial node embedding (5) $\mathbf { x } _ { i } ^ { ( 0 ) } =$ $( \phi ( x _ { i } ) , h _ { i } )$ , using a non-trainable edge feature
|
| 130 |
+
|
| 131 |
+
$$
|
| 132 |
+
\varphi ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) = \lVert \phi ( x _ { i } ) - \phi ( x _ { j } ) \rVert , \tilde { A } ^ { ( 0 ) } = \mathrm { s o f t m a x } ( - \varphi ) ,
|
| 133 |
+
$$
|
| 134 |
+
|
| 135 |
+
and resulting label estimation
|
| 136 |
+
|
| 137 |
+
$$
|
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+
\hat { Y } _ { * } = \sum _ { j } \tilde { A } _ { * , j } ^ { ( 0 ) } \langle \mathbf { x } _ { j } ^ { ( 0 ) } , u \rangle ,
|
| 139 |
+
$$
|
| 140 |
+
|
| 141 |
+
with $u$ selecting the label field from $\mathbf { x }$ . In this model, the learning is reduced to learning image embeddings $\phi ( x _ { i } )$ whose euclidean metric is consistent with the label similarities.
|
| 142 |
+
|
| 143 |
+
Prototypical Networks Prototypical networks Snell et al. (2017) evolve Siamese networks by aggregating information within each cluster determined by nodes with the same label. This operation can also be accomplished with a gnn as follows. we consider
|
| 144 |
+
|
| 145 |
+
$$
|
| 146 |
+
\tilde { A } _ { i , j } ^ { ( 0 ) } = \left\{ \begin{array} { c c } { q ^ { - 1 } } & { \mathrm { i f } l _ { i } = l _ { j } } \\ { 0 } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
|
| 147 |
+
$$
|
| 148 |
+
|
| 149 |
+
where $q$ is the number of examples per class, and
|
| 150 |
+
|
| 151 |
+
$$
|
| 152 |
+
\mathbf { x } _ { i } ^ { ( 1 ) } = \sum _ { j } \tilde { A } _ { i , j } ^ { ( 0 ) } \mathbf { x } _ { j } ^ { ( 0 ) } ,
|
| 153 |
+
$$
|
| 154 |
+
|
| 155 |
+
where $\mathbf { x } ^ { ( 0 ) }$ is defined as in the Siamese Networks. We finally apply the previous kernel $\tilde { A } ^ { ( 1 ) } =$ softmax $\left( \varphi \right)$ applied to $\mathbf { x } ^ { ( 1 ) }$ to yield class prototypes:
|
| 156 |
+
|
| 157 |
+
$$
|
| 158 |
+
\hat { Y } _ { * } = \sum _ { j } \tilde { A } _ { * , j } ^ { ( 1 ) } \langle \mathbf { x } _ { j } ^ { ( 1 ) } , u \rangle .
|
| 159 |
+
$$
|
| 160 |
+
|
| 161 |
+
Matching Networks Matching networks Vinyals et al. (2016) use a set representation for the ensemble of images in $\tau$ , similarly as our proposed graph neural network model, but with two important differences. First, the attention mechanism considered in this set representation is akin to the edge feature learning, with the difference that the mechanism attends always to the same node embeddings, as opposed to our stacked adjacency learning, which is closer to Vaswani et al. (2017). In other words, instead of the attention kernel in (3), matching networks consider attention mechanisms of the form A˜(k)∗,j $\tilde { A } _ { * , j } ^ { ( k ) } = \varphi ( \mathbf { x } _ { * } ^ { ( k ) } , \mathbf { x } _ { j } ^ { ( T ) } )$ , where $\underset { - } { \mathbf { x } _ { j } ^ { ( T ) } }$ is the encoding function for the elements of the support set, obtained with bidirectional LSTMs. In that case, the support set encoding is thus computed independently of the target image. Second, the label and image fields are treated separately throughout the model, with a final step that aggregates linearly the labels using a trained kernel. This may prevent the model to leverage complex dependencies between labels and images at intermediate stages.
|
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+
|
| 163 |
+
# 5 TRAINING
|
| 164 |
+
|
| 165 |
+
We describe next how to train the parameters of the GNN in the different setups we consider: fewshot learning, semi-supervised learning and active learning.
|
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+
|
| 167 |
+
# 5.1 FEW-SHOT AND SEMI-SUPERVISED LEARNING
|
| 168 |
+
|
| 169 |
+
In this setup, the model is asked only to predict the label $Y$ corresponding to the image to classify $\bar { x } \in \tau$ , associated with node $^ *$ in the graph. The final layer of the GNN is thus a softmax mapping the node features to the $K$ -simplex. We then consider the Cross-entropy loss evaluated at node $^ *$ :
|
| 170 |
+
|
| 171 |
+
$$
|
| 172 |
+
\ell ( \Phi ( \mathcal { T } ; \Theta ) , Y ) = - \sum _ { k } y _ { k } \log P ( Y _ { * } = y _ { k } \mid \mathcal { T } ) .
|
| 173 |
+
$$
|
| 174 |
+
|
| 175 |
+
The semi-supervised setting is trained identically — the only difference is that the initial label fields of the node will be filled with the uniform distribution on nodes corresponding to $\tilde { x } _ { j }$ .
|
| 176 |
+
|
| 177 |
+
# 5.2 ACTIVE LEARNING
|
| 178 |
+
|
| 179 |
+
In the Active Learning setup, the model has the intrinsic ability to query for one of the labels from $\{ \tilde { x } _ { 1 } , \ldots , \tilde { x } _ { r } \}$ . The network will learn to ask for the most informative label in order to classify the sample $\bar { x } \in \mathcal { T }$ . The querying is done after the first layer of the GNN by using a Softmax attention over the unlabeled nodes of the graph. For this we apply a function $g ( \mathbf { x } _ { i } ^ { ( 1 ) } ) \in \mathbb { R } ^ { 1 }$ that maps each unlabeled vector node to a scalar value. Function $g$ is parametrized by a two layers neural network. A Softmax is applied over the $\{ 1 , \ldots , r \}$ scalar values obtained after applying $g$ :
|
| 180 |
+
|
| 181 |
+
$$
|
| 182 |
+
\mathrm { A t t e n t i o n } = \mathrm { S o f t m a x } ( g ( \mathbf { x } _ { \{ 1 , \dots , r \} } ^ { ( 1 ) } ) )
|
| 183 |
+
$$
|
| 184 |
+
|
| 185 |
+
In order to query only one sample, we set all elements from the $A t t e n t i o n \in \mathbb { R } ^ { r }$ vector to 0 except for one. At test time we keep the maximum value, at train time we randomly sample one value based on its multinomial probability. Then we multiply this sampled attention by the label vectors:
|
| 186 |
+
|
| 187 |
+
$$
|
| 188 |
+
w \cdot h ( l _ { i ^ { * } } ) = \langle \mathrm { A t t e n t i o n } ^ { \prime } , h ( l _ { \{ 1 , \dots , r \} } ) \rangle
|
| 189 |
+
$$
|
| 190 |
+
|
| 191 |
+
The label of the queried vector $h ( l _ { i ^ { * } } )$ is obtained, scaled by the weight $w \in ( 0 , 1 )$ . This value is then summed to the current representation $\mathbf { x } _ { i ^ { * } } ^ { ( 1 ) }$ , since we are using dense connections in our GNN model we can sum this $w \cdot h ( l _ { i ^ { * } } )$ value directly to where the uniform label distribution was concatenated
|
| 192 |
+
|
| 193 |
+
$$
|
| 194 |
+
\mathbf { x } _ { i ^ { * } } ^ { ( 1 ) } = [ \mathbf { G c } ( \mathbf { x } _ { i ^ { * } } ^ { ( 0 ) } ) , \mathbf { x } _ { i ^ { * } } ^ { ( 0 ) } ] = [ \mathbf { G c } ( \mathbf { x } _ { i ^ { * } } ^ { ( 0 ) } ) , ( \phi ( x _ { i ^ { * } } ) , h ( l _ { i ^ { * } } ) ) ]
|
| 195 |
+
$$
|
| 196 |
+
|
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After the label has been summed to the current node, the information is forward propagated. This attention part is trained end-to-end with the rest of the network by backpropagating the loss from the output of the GNN.
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# 6 EXPERIMENTS
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For the few-shot, semi-supervised and active learning experiments we used the Omniglot dataset presented by Lake et al. (2015) and Mini-Imagenet dataset introduced by Vinyals et al. (2016) which is a small version of ILSVRC-12 Krizhevsky et al. (2012). All experiments are based on the $q$ -shot, $K$ -way setting. For all experiments we used the same values $q$ -shot and $K$ -way for both training and testing.
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Code available at: https://github.com/vgsatorras/few-shot-gnn
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# 6.1 DATASETS AND IMPLEMENTATION
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# 6.1.1 OMNIGLOT
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Dataset: Omniglot is a dataset of 1623 characters from 50 different alphabets, each character/class has been drawn by 20 different people. Following Vinyals et al. (2016) implementation we split the dataset into 1200 classes for training and the remaining 423 for testing. We augmented the dataset by multiples of 90 degrees as proposed by Santoro et al. (2016).
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Architectures: Inspired by the embedding architecture from Vinyals et al. (2016), following Mishra et al. (2017), a CNN was used as an embedding $\phi$ function consisting of four stacked blocks of $\{ 3 \times 3$ -convolutional layer with 64 filters, batch-normalization, $2 \times 2$ max-pooling, leaky-relu} the output is passed through a fully connected layer resulting in a 64-dimensional embedding. For the GNN we used 3 blocks each of them composed by 1) a module that computes the adjacency matrix and 2) a graph convolutional layer. A more detailed description of each block can be found at Figure 3.
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# 6.1.2 MINI-IMAGENET
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Dataset: Mini-Imagenet is a more challenging dataset for one-shot learning proposed by Vinyals et al. (2016) derived from the original ILSVRC-12 dataset Krizhevsky et al. (2012). It consists of $8 4 \times 8 4$ RGB images from 100 different classes with 600 samples per class. It was created with the purpose of increasing the complexity for one-shot tasks while keeping the simplicity of a light size dataset, that makes it suitable for fast prototyping. We used the splits proposed by Ravi & Larochelle (2016) of 64 classes for training, 16 for validation and 20 for testing. Using 64 classes for training, and the 16 validation classes only for early stopping and parameter tuning.
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Architecture: The embedding architecture used for Mini-Imagenet is formed by 4 convolutional layers followed by a fully-connected layer resulting in a 128 dimensional embedding. This light architecture is useful for fast prototyping:
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$1 \times \{ 3 \times 3$ -conv. layer (64 filters), batch normalization, max pool $( 2 , 2 )$ , leaky relu $\}$ , $1 \times \{ 3 \times 3$ -conv. layer (96 filters), batch normalization, max $\mathrm { p o o l } ( 2 , 2 )$ , leaky relu}, $1 \times \{ 3 \times 3$ -conv. layer (128 filters), batch normalization, max $\mathrm { p o o l } ( 2 , 2 )$ , leaky relu, dropout $\left( 0 . 5 \right) \}$ , $1 \times \{ 3 \times 3$ -conv. layer (256 filters), batch normalization, max $\mathrm { p o o l } ( 2 , 2 )$ , leaky relu, dropout $\left( 0 . 5 \right) \}$ , $1 \times \left\{ \begin{array} { r l } \end{array} \right.$ fc-layer (128 filters), batch normalization}.
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The two dropout layers are useful to avoid overfitting the GNN in Mini-Imagenet dataset. The GNN architecture is similar than for Omniglot, it is formed by 3 blocks, each block is described at Figure 3.
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# 6.2 FEW-SHOT
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Few-shot learning experiments for Omniglot and Mini-Imagenet are presented at Table 1 and Table 2 respectively.
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We evaluate our model by performing different ${ \bf q }$ -shot, K-way experiments on both datasets. For every few-shot task $\tau$ , we sample $K$ random classes from the dataset, and from each class we sample $q$ random samples. An extra sample to classify is chosen from one of that $K$ classes.
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Omniglot: The GNN method is providing competitive results while still remaining simpler than other methods. State of the art results are reached in the 5-Way and 20-way 1-shot experiments. In the 20-Way 1-shot setting the GNN is providing slightly better results than Munkhdalai & Yu (2017) while still being a more simple approach. The TCML approach from Mishra et al. (2017) is in the same confidence interval for 3 out of 4 experiments, but it is slightly better for the 20-Way 5-shot, although the number of parameters is reduced from ${ \sim } 5 \mathbf { M }$ (TCML) to $\sim 3 0 0 \mathrm { K }$ (3 layers GNN).
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At Mini-Imagenet table we are also presenting a baseline ”Our metric learning $\mathbf { \chi } + K N N ^ { \prime }$ where no information has been aggregated among nodes, it is a K-nearest neighbors applied on top of the pair-wise learnable metric $\bar { \varphi _ { \theta } } ( \mathbf { x } _ { i } ^ { ( 0 ) } , \mathbf { x } _ { j } ^ { ( 0 ) } )$ and trained end-to-end, this learnable metric is competitive by itself compared to other state of the art methods. Even so, a significant improvement (from $6 4 . 0 2 \%$ to $6 6 . 4 1 \%$ ) can be seen for the 5-shot 5-Way Mini-Imagenet setting when aggregating information among nodes by using the full GNN architecture. A variety of embedding functions $\phi$ are used among the different papers for Mini-Imagenet experiments, in our case we are using a simple network of 4 conv. layers followed by a fully connected layer (Section 6.1.2) which served us to compare between Our GNN and Our metric learning $+ ~ K N N$ and it is useful for fast prototyping. More complex embeddings have proven to produce better results, at Mishra et al. (2017) a deep residual network is used as embedding network $\phi$ increasing the accuracy considerably. Regarding the TCML architecture in Mini-Imagenet, the number of parameters is reduced from ${ \sim } 1 1 \mathbf { M }$ (TCML) to ${ \sim } 4 0 0 \mathrm { K }$ (3 layers GNN).
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<table><tr><td rowspan="2"></td><td colspan="2"> 5-Way</td><td colspan="2">20-Way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Model Pixels Vinyals et al. (2016)</td><td>41.7%</td><td>63.2%</td><td>26.7%</td><td>42.6%</td></tr><tr><td>Siamese Net Koch et al. (2015)</td><td>97.3%</td><td>98.4%</td><td>88.2%</td><td>97.0%</td></tr><tr><td>Matching Networks Vinyals et al. (2016)</td><td>98.1%</td><td>98.9%</td><td>93.8%</td><td>98.5%</td></tr><tr><td>N.Statistician Edwards & Storkey (2016)</td><td>98.1%</td><td>99.5%</td><td>93.2%</td><td>98.1%</td></tr><tr><td>Res.Pair-Wise Mehrotra & Dukkipati (2017)</td><td>-</td><td>-</td><td>94.8%</td><td>-</td></tr><tr><td>Prototypical Networks Snellet al. (2017)</td><td>97.4%</td><td>99.3%</td><td>95.4%</td><td>98.8%</td></tr><tr><td>ConvNet with Memory Kaiser et al. (2017)</td><td>98.4%</td><td>99.6%</td><td>95.0%</td><td>98.6%</td></tr><tr><td>Agnostic Meta-learner Finn et al. (2017)</td><td>98.7 ±0.4%</td><td>99.9 ±0.3%</td><td>95.8 ±0.3%</td><td>98.9 ±0.2%</td></tr><tr><td>MetaNetworks Munkhdalai& Yu (2017)</td><td>98.9%</td><td>=</td><td>97.0%</td><td></td></tr><tr><td>TCML Mishra et al. (2017)</td><td></td><td>98.96% ±0.20% 99.75% ±0.11% 97.64% ±0.30% 99.36% ±0.18%</td><td></td><td></td></tr><tr><td>Our GNN</td><td>99.2%</td><td>99.7%</td><td>97.4%</td><td>99.0%</td></tr></table>
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Table 1: Few-Shot Learning — Omniglot accuracies. Siamese Net results are extracted from Vinyals et al. (2016) reimplementation.
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Table 2: Few-shot learning — Mini-Imagenet average accuracies with $9 5 \%$ confidence intervals.
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<table><tr><td rowspan="2">Model</td><td colspan="2">5-Way</td></tr><tr><td>1-shot</td><td>5-shot</td></tr><tr><td>Matching Networks Vinyals et al. (2016)</td><td>43.6%</td><td>55.3%</td></tr><tr><td>Prototypical Networks Snel et al. (2017)</td><td>46.61% ±0.78%</td><td>65.77% ±0.70%</td></tr><tr><td>Model Agnostic Meta-learner Finn et al. (2017)</td><td>48.70% ±1.84%</td><td>63.1% ±0.92%</td></tr><tr><td>Meta Networks Munkhdalai & Yu (2017)</td><td>49.21% ±0.96</td><td></td></tr><tr><td>Ravi &Larochelle Ravi & Larochelle (2016)</td><td>43.4% ±0.77%</td><td>60.2% ±0.71%</td></tr><tr><td>TCML Mishra et al. (2017)</td><td>55.71% ±0.99%</td><td>68.88% ±0.92%</td></tr><tr><td>Our metric learning + KNN</td><td>49.44% ±0.28%</td><td>64.02% ±0.51%</td></tr><tr><td>Our GNN</td><td>50.33% ±0.36%</td><td>66.41% ±0.63%</td></tr></table>
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# 6.3 SEMI-SUPERVISED
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Semi-supervised experiments are performed on the 5-way 5-shot setting. Different results are presented when $20 \%$ and $40 \%$ of the samples are labeled. The labeled samples are balanced among classes in all experiments, in other words, all the classes have the same amount of labeled and unlabeled samples.
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Two strategies can be seen at Tables 3 and 4. ”GNN - Trained only with labeled” is equivalent to the supervised few-shot setting, for example, in the 5-Way 5-shot $20 \%$ -labeled setting, this method is equivalent to the 5-way 1-shot learning setting since it is ignoring the unlabeled samples. ”GNN - Semi supervised” is the actual semi-supervised method, for example, in the 5-Way 5-shot $20 \%$ - labeled setting, the GNN receives as input 1 labeled sample per class and 4 unlabeled samples per class.
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Omniglot results are presented at Table 3, for this scenario we observe that the accuracy improvement is similar when adding images than when adding labels. The GNN is able to extract information from the input distribution of unlabeled samples such that only using $20 \%$ of the labels in a 5-shot semi-supervised environment we get same results as in the $40 \%$ supervised setting.
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In Mini-Imagenet experiments, Table 4, we also notice an improvement when using semi-supervised data although it is not as significant as in Omniglot. The distribution of Mini-Imagenet images is more complex than for Omniglot. In spite of it, the GNN manages to improve by ${ \sim } 2 \%$ in the $20 \%$ and $40 \%$ settings.
|
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Table 3: Semi-Supervised Learning — Omniglot accuracies.
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<table><tr><td></td><td colspan="3">5-Way 5-shot</td></tr><tr><td>Model</td><td>20%-labeled</td><td>40%-labeled</td><td>100 %-labeled</td></tr><tr><td>GNN - Trained only with labeled</td><td>99.18%</td><td>99.59%</td><td>99.71%</td></tr><tr><td>GNN -Semi supervised</td><td>99.59%</td><td>99.63%</td><td>99.71%</td></tr></table>
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Table 4: Semi-Supervised Learning — Mini-Imagenet average accuracies with $9 5 \%$ confidence intervals.
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<table><tr><td></td><td colspan="3">5-Way 5-shot</td></tr><tr><td>Model</td><td>20%-labeled</td><td>40%-labeled</td><td>100 %-labeled</td></tr><tr><td>GNN - Trainedonlywithlabeled</td><td>50.33% ±0.36%</td><td>56.91% ±0.42%</td><td>66.41% ±0.63%</td></tr><tr><td>GNN - Semi supervised</td><td>52.45% ±0.88%</td><td>58.76% ±0.86%</td><td>66.41% ±0.63%</td></tr></table>
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# 6.4 ACTIVE LEARNING
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We performed Active Learning experiments on the 5-Way 5-shot set-up when $20 \%$ of the samples are labeled. In this scenario our network will query for the label of one sample from the unlabeled ones. The results are compared with the Random baseline where the network chooses a random sample to be labeled instead of one that maximally reduces the loss of the classification task $\tau$ .
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Results are shown at Table 5. The results of the GNN-Random criterion are close to the Semisupervised results for $20 \%$ -labeled samples from Tables 3 and 4. It means that selecting one random label practically does not improve the accuracy at all. When using the GNN-AL learned criterion, we notice an improvement of $\sim 3 . 4 \%$ for Mini-Imagenet, it means that the GNN manages to correctly choose a more informative sample than a random one. In Omniglot the improvement is smaller since the accuracy is almost saturated and the improving margin is less.
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<table><tr><td>Method</td><td>5-Way 5-shot20%-labeled</td><td>Method</td><td>5-Way 5-shot 20%-labeled</td></tr><tr><td>GNN - AL</td><td>99.62%</td><td>GNN - AL</td><td>55.99% ±1.35%</td></tr><tr><td>GNN - Random</td><td>99.59%</td><td>GNN - Random</td><td>52.56% ±1.18%</td></tr></table>
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Table 5: Omniglot (left) and Mini-Imagneet (right), average accuracies are shown at both tables, the GNN-AL is the learned criterion that performs Active Learning by selecting the sample that will maximally reduce the loss of the current classification. The GNN - Random is also selecting one sample, but in this case a random one. Mini-Imagenet results are presented with $9 5 \%$ confidence intervals.
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# 7 CONCLUSIONS
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This paper explored graph neural representations for few-shot, semi-supervised and active learning. From the meta-learning perspective, these tasks become supervised learning problems where the input is given by a collection or set of elements, whose relational structure can be leveraged with neural message passing models. In particular, stacked node and edge features generalize the contextual similarity learning underpinning previous few-shot learning models.
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The graph formulation is helpful to unify several training setups (few-shot, active, semi-supervised) under the same framework, a necessary step towards the goal of having a single learner which is able to operate simultaneously in different regimes (stream of labels with few examples per class, or stream of examples with few labels). This general goal requires scaling up graph models to millions of nodes, motivating graph hierarchical and coarsening approaches Defferrard et al. (2016).
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Another future direction is to generalize the scope of Active Learning, to include e.g. the ability to ask questions Rothe et al. (2017), or in reinforcement learning setups, where few-shot learning is critical to adapt to non-stationary environments.
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# ACKNOWLEDGMENTS
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This work was partly supported by Samsung Electronics (Improving Deep Learning using Latent Structure).
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# APPENDIX
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Figure 3: GNN model. Three blue blocks are used for Omniglot and Mini-Imagenet. $( \mathrm { n f } { = } 9 6 ) ,$ ).
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "FEW-SHOT LEARNING WITH GRAPH NEURAL NETWORKS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
+
174,
|
| 8 |
+
99,
|
| 9 |
+
821,
|
| 10 |
+
145
|
| 11 |
+
],
|
| 12 |
+
"page_idx": 0
|
| 13 |
+
},
|
| 14 |
+
{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Victor Garcia∗ Amsterdam Machine Learning Lab University of Amsterdam Amsterdam, 1098 XH, NL v.garciasatorras@uva.nl ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
184,
|
| 19 |
+
171,
|
| 20 |
+
415,
|
| 21 |
+
239
|
| 22 |
+
],
|
| 23 |
+
"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Joan Bruna \nCourant Institute of Mathematical Sciences \nNew York University \nNew York City, NY, 10010, USA \nbruna@cims.nyu.edu ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
480,
|
| 30 |
+
171,
|
| 31 |
+
766,
|
| 32 |
+
239
|
| 33 |
+
],
|
| 34 |
+
"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "ABSTRACT ",
|
| 39 |
+
"text_level": 1,
|
| 40 |
+
"bbox": [
|
| 41 |
+
454,
|
| 42 |
+
276,
|
| 43 |
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544,
|
| 44 |
+
291
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| 45 |
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],
|
| 46 |
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"page_idx": 0
|
| 47 |
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},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "We propose to study the problem of few-shot learning with the prism of inference on a partially observed graphical model, constructed from a collection of input images whose label can be either observed or not. By assimilating generic message-passing inference algorithms with their neural-network counterparts, we define a graph neural network architecture that generalizes several of the recently proposed few-shot learning models. Besides providing improved numerical performance, our framework is easily extended to variants of few-shot learning, such as semi-supervised or active learning, demonstrating the ability of graph-based models to operate well on ‘relational’ tasks. ",
|
| 51 |
+
"bbox": [
|
| 52 |
+
233,
|
| 53 |
+
306,
|
| 54 |
+
764,
|
| 55 |
+
430
|
| 56 |
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],
|
| 57 |
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"page_idx": 0
|
| 58 |
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},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "1 INTRODUCTION ",
|
| 62 |
+
"text_level": 1,
|
| 63 |
+
"bbox": [
|
| 64 |
+
176,
|
| 65 |
+
454,
|
| 66 |
+
336,
|
| 67 |
+
469
|
| 68 |
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],
|
| 69 |
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"page_idx": 0
|
| 70 |
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},
|
| 71 |
+
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"text": "Supervised end-to-end learning has been extremely successful in computer vision, speech, or machine translation tasks, thanks to improvements in optimization technology, larger datasets and streamlined designs of deep convolutional or recurrent architectures. Despite these successes, this learning setup does not cover many aspects where learning is nonetheless possible and desirable. ",
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"text": "One such instance is the ability to learn from few examples, in the so-called few-shot learning tasks. Rather than relying on regularization to compensate for the lack of data, researchers have explored ways to leverage a distribution of similar tasks, inspired by human learning Lake et al. (2015). This defines a new supervised learning setup (also called ‘meta-learning’) in which the input-output pairs are no longer given by iid samples of images and their associated labels, but by iid samples of collections of images and their associated label similarity. ",
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"text": "A recent and highly-successful research program has exploited this meta-learning paradigm on the few-shot image classification task Lake et al. (2015); Koch et al. (2015); Vinyals et al. (2016); Mishra et al. (2017); Snell et al. (2017). In essence, these works learn a contextual, task-specific similarity measure, that first embeds input images using a CNN, and then learns how to combine the embedded images in the collection to propagate the label information towards the target image. ",
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"text": "In particular, Vinyals et al. (2016) cast the few-shot learning problem as a supervised classification task mapping a support set of images into the desired label, and developed an end-to-end architecture accepting those support sets as input via attention mechanisms. In this work, we build upon this line of work, and argue that this task is naturally expressed as a supervised interpolation problem on a graph, where nodes are associated with the images in the collection, and edges are given by a trainable similarity kernels. Leveraging recent progress on representation learning for graphstructured data Bronstein et al. (2017); Gilmer et al. (2017), we thus propose a simple graph-based few-shot learning model that implements a task-driven message passing algorithm. The resulting architecture is trained end-to-end, captures the invariances of the task, such as permutations within the input collections, and offers a good tradeoff between simplicity, generality, performance and sample complexity. ",
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"text": "Besides few-shot learning, a related task is the ability to learn from a mixture of labeled and unlabeled examples — semi-supervised learning, as well as active learning, in which the learner has the option to request those missing labels that will be most helpful for the prediction task. Our graphbased architecture is naturally extended to these setups with minimal changes in the training design. We validate experimentally the model on few-shot image classification, matching state-of-the-art performance with considerably fewer parameters, and demonstrate applications to semi-supervised and active learning setups. ",
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"text": "",
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"text": "Our contributions are summarized as follows: ",
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"text": "• We cast few-shot learning as a supervised message passing task which is trained end-to-end using graph neural networks. \n• We match state-of-the-art performance on Omniglot and Mini-Imagenet tasks with fewer parameters. \n• We extend the model in the semi-supervised and active learning regimes. ",
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"text": "The rest of the paper is structured as follows. Section 2 describes related work, Sections 3, 4 and 5 present the problem setup, our graph neural network model and the training, and Section 6 reports numerical experiments. ",
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"text": "2 RELATED WORK ",
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"text": "One-shot learning was first introduced by Fei-Fei et al. (2006), they assumed that currently learned classes can help to make predictions on new ones when just one or few labels are available. More recently, Lake et al. (2015) presented a Hierarchical Bayesian model that reached human level error on few-shot learning alphabet recongition tasks. ",
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"text": "Since then, great progress has been done in one-shot learning. Koch et al. (2015) presented a deeplearning model based on computing the pair-wise distance between samples using Siamese Networks, then, this learned distance can be used to solve one-shot problems by $\\mathbf { k }$ -nearest neighbors classification. Vinyals et al. (2016) Presented an end-to-end trainable $\\mathbf { k }$ -nearest neighbors using the cosine distance, they also introduced a contextual mechanism using an attention LSTM model that takes into account all the samples of the subset $\\tau$ when computing the pair-wise distance between samples. Snell et al. (2017) extended the work from Vinyals et al. (2016), by using euclidean distance instead of cosine which provided significant improvements, they also build a prototype representation of each class for the few-shot learning scenario. Mehrotra & Dukkipati (2017) trained a deep residual network together with a generative model to approximate the pair-wise distance between samples. ",
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"text": "A new line of meta-learners for one-shot learning is rising lately: Ravi & Larochelle (2016) introduced a meta-learning method where an LSTM updates the weights of a classifier for a given episode. Munkhdalai & Yu (2017) also presented a meta-learning architecture that learns meta-level knowledge across tasks, and it changes its inductive bias via fast parametrization. Finn et al. (2017) is using a model agnostic meta-learner based on gradient descent, the goal is to train a classification model such that given a new task, a small amount of gradient steps with few data will be enough to generalize. Lately, Mishra et al. (2017) used Temporal Convolutions which are deep recurrent networks based on dilated convolutions, this method also exploits contextual information from the subset $\\tau$ providing very good results. ",
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"type": "text",
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"text": "Another related area of research concerns deep learning architectures on graph-structured data. The GNN was first proposed in Gori et al. (2005); Scarselli et al. (2009), as a trainable recurrent messagepassing whose fixed points could be adjusted discriminatively. Subsequent works Li et al. (2015); Sukhbaatar et al. (2016) have relaxed the model by untying the recurrent layer weights and proposed several nonlinear updates through gating mechanisms. Graph neural networks are in fact natural generalizations of convolutional networks to non-Euclidean graphs. Bruna et al. (2013); Henaff et al. (2015) proposed to learn smooth spectral multipliers of the graph Laplacian, albeit with high computational cost, and Defferrard et al. (2016); Kipf & Welling (2016) resolved the computational bottleneck by learning polynomials of the graph Laplacian, thus avoiding the computation of eigenvectors and completing the connection with GNNs. In particular, Kipf & Welling (2016) was the first to propose the use of GNNs on semi-supervised classification problems. We refer the reader to Bronstein et al. (2017) for an exhaustive literature review on the topic. GNNs and the analogous Neural Message Passing Models are finding application in many different domains. Battaglia et al. ",
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"text": "(2016); Chang et al. (2016) develop graph interaction networks that learn pairwise particle interactions and apply them to discrete particle physical dynamics. Duvenaud et al. (2015); Kearnes et al. (2016) study molecular fingerprints using variants of the GNN architecture, and Gilmer et al. (2017) further develop the model by combining it with set representations Vinyals et al. (2015), showing state-of-the-art results on molecular prediction. ",
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"type": "text",
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"text": "3 PROBLEM SET-UP ",
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"text": "We describe first the general setup and notations, and then particularize it to the case of few-shot learning, semi-supervised learning and active learning. ",
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"text": "We consider input-output pairs $( \\mathcal { T } _ { i } , Y _ { i } ) _ { i }$ drawn iid from a distribution $P$ of partially-labeled image collections ",
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"type": "equation",
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"img_path": "images/8251fc0b282a836b4bb81b5fc5257e1baa263762d8814589e46d779e1a1c488e.jpg",
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"text": "$$\n\\begin{array} { r c l } { { { \\cal T } } } & { { = } } & { { \\{ \\{ ( x _ { 1 } , l _ { 1 } ) , \\dots ( x _ { s } , l _ { s } ) \\} , \\{ \\tilde { x } _ { 1 } , \\dots , \\tilde { x } _ { r } \\} , \\{ \\bar { x } _ { 1 } , \\dots , \\bar { x } _ { t } \\} ; l _ { i } \\in \\{ 1 , K \\} , x _ { i } , \\tilde { x } _ { j } , \\bar { x } _ { j } \\sim \\{ \\mathcal { P } _ { l } ( \\mathbb { R } ^ { N } ) \\} \\} , } } \\\\ { { { \\cal Y } } } & { { = } } & { { ( y _ { 1 } , \\dots , y _ { t } ) \\in \\{ 1 , K \\} ^ { t } , } } \\\\ { { } } & { { } } & { { ( 1 ) } } \\end{array}\n$$",
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"text": "for arbitrary values of $s , r , t$ and $K$ . Where $s$ is the number of labeled samples, $r$ is the number of unlabeled samples $r > 0$ for the semi-supervised and active learning scenarios) and $t$ is the number of samples to classify. $K$ is the number of classes. We will focus in the case $t = 1$ where we just classify one sample per task $\\tau$ . $\\mathcal { P } _ { l } ( \\mathbb { R } ^ { N } )$ denotes a class-specific image distribution over $\\mathbb { R } ^ { N }$ . In our context, the targets $Y _ { i }$ are associated with image categories of designated images $\\bar { x } _ { 1 } , \\ldots , \\bar { x } _ { t } \\in \\mathcal { T } _ { i }$ with no observed label. Given a training set $\\{ ( \\mathcal { T } _ { i } , Y _ { i } ) _ { i } \\} _ { i \\leq L }$ , we consider the standard supervised learning objective ",
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"text": "$$\n\\operatorname* { m i n } _ { \\Theta } \\frac { 1 } { L } \\sum _ { i \\leq L } \\ell ( \\Phi ( \\mathcal { T } _ { i } ; \\Theta ) , Y _ { i } ) + \\mathcal { R } ( \\Theta ) ,\n$$",
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"text": "using the model $\\Phi ( { \\mathcal { T } } ; \\Theta ) = p ( Y \\mid { \\mathcal { T } } )$ specified in Section 4 and $\\mathcal { R }$ is a standard regularization objective. ",
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"text": "Few-Shot Learning When $r = 0$ , $t = 1$ and $s = q K$ , there is a single image in the collection with unknown label. If moreover each label appears exactly $q$ times, this setting is referred as the $q$ -shot, $K$ -way learning. ",
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"text": "Semi-Supervised Learning When $r > 0$ and $t = 1$ , the input collection contains auxiliary images $\\tilde { x } _ { 1 } , \\ldots , \\tilde { x } _ { r }$ that the model can use to improve the prediction accuracy, by leveraging the fact that these samples are drawn from common distributions as those determining the output. ",
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"text": "Active Learning In the active learning setting, the learner has the ability to request labels from the sub-collection $\\{ \\tilde { x } _ { 1 } , \\ldots , \\tilde { x } _ { r } \\}$ . We are interested in studying to what extent this active learning can improve the performance with respect to the previous semi-supervised setup, and match the performance of the one-shot learning setting with $s _ { 0 }$ known labels when $s + r = s _ { 0 }$ , $s \\ll s _ { 0 }$ . ",
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"text": "4 MODEL ",
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"text": "This section presents our approach, based on a simple end-to-end graph neural network architecture. We first explain how the input context is mapped into a graphical representation, then detail the architecture, and next show how this model generalizes a number of previously published few-shot learning architectures. ",
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"text": "4.1 SET AND GRAPH INPUT REPRESENTATIONS ",
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"text": "The input $\\tau$ contains a collection of images, both labeled and unlabeled. The goal of few-shot learning is to propagate label information from labeled samples towards the unlabeled query image. This propagation of information can be formalized as a posterior inference over a graphical model determined by the input images and labels. ",
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"text": "Following several recent works that cast posterior inference using message passing with neural networks defined over graphs Scarselli et al. (2009); Duvenaud et al. (2015); Gilmer et al. (2017), we associate $\\tau$ with a fully-connected graph $G _ { \\mathcal { T } } = ( V , E )$ where nodes $v _ { a } \\in V$ correspond to the images present in $\\tau$ (both labeled and unlabeled). In this context, the setup does not specify a fixed similarity $e _ { a , a ^ { \\prime } }$ between images $x _ { a }$ and $x _ { a ^ { \\prime } }$ , suggesting an approach where this similarity measure is learnt in a discriminative fashion with a parametric model similarly as in Gilmer et al. (2017), such as a siamese neural architecture. This framework is closely related to the set representation from Vinyals et al. (2016), but extends the inference mechanism using the graph neural network formalism that we detail next. ",
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"img_path": "images/f38eabf35cdc750fbe4969ce4ddaefa5ad674cad5880cb8253ae0c611d3881fd.jpg",
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"image_caption": [
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| 413 |
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"Figure 1: Visual representation of One-Shot Learning setting. "
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"type": "text",
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"text": "4.2 GRAPH NEURAL NETWORKS ",
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"text": "Graph Neural Networks, introduced in Gori et al. (2005); Scarselli et al. (2009) and further simplified in Li et al. (2015); Duvenaud et al. (2015); Sukhbaatar et al. (2016) are neural networks based on local operators of a graph $G = ( V , E )$ , offering a powerful balance between expressivity and sample complexity; see Bronstein et al. (2017) for a recent survey on models and applications of deep learning on graphs. ",
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"text": "In its simplest incarnation, given an input signal $F \\in \\mathbb { R } ^ { V \\times d }$ on the vertices of a weighted graph $G$ , we consider a family $\\mathcal { A }$ of graph intrinsic linear operators that act locally on this signal. The simplest is the adjacency operator $\\mathsf { \\bar { A } } : \\mathsf { F } \\mapsto A ( F )$ where $\\begin{array} { r } { ( A F ) _ { i } : = \\sum _ { j \\sim i } w _ { i , j } \\mathbf { \\bar { F } } _ { j } } \\end{array}$ , with $i \\sim j$ iff $( i , j ) \\in E$ and $w _ { i , j }$ its associated weight. A GNN layer $\\operatorname { G c } ( \\cdot )$ receives as input a signal $\\mathbf { x } ^ { ( k ) } \\in \\mathbb { R } ^ { V \\times d _ { k } }$ and produces $\\mathbf { x } ^ { ( k + 1 ) } \\in \\mathbb { R } ^ { V \\times d _ { k + 1 } }$ as ",
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"text": "$$\n{ \\bf x } _ { l } ^ { ( k + 1 ) } = { \\bf G } { \\bf c } ( { \\bf x } ^ { ( k ) } ) = \\rho \\left( \\sum _ { B \\in \\mathcal { A } } B { \\bf x } ^ { ( k ) } \\theta _ { B , l } ^ { ( k ) } \\right) \\mathrm { ~ , ~ } l = d _ { 1 } \\mathrm { ~ . ~ . ~ } d _ { k + 1 } \\mathrm { ~ , ~ }\n$$",
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"text": "where $\\Theta = \\{ \\theta _ { 1 } ^ { ( k ) } , \\dots , \\theta _ { | \\mathcal { A } | } ^ { ( k ) } \\} _ { k }$ , ${ \\boldsymbol { \\theta } _ { B } ^ { ( k ) } \\in \\mathbb { R } ^ { d _ { k } \\times d _ { k + 1 } } }$ , are trainable parameters and $\\rho ( \\cdot )$ is a point-wise non-linearity, chosen in this work to be a ‘leaky’ ReLU et al. (2015). ",
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"text": "Authors have explored several modeling variants from this basic formulation, by replacing the pointwise nonlinearity with gating operations Duvenaud et al. (2015), or by generalizing the generator family to Laplacian polynomials Defferrard et al. (2016); Kipf & Welling (2016); Bruna et al. (2013), or including $2 ^ { J }$ -th powers of $A$ to $\\mathcal { A }$ , $A _ { J } = \\operatorname* { m i n } ( 1 , A ^ { 2 ^ { J } } )$ to encode $2 ^ { J }$ -hop neighborhoods of each node Bruna & Li (2017). Cascaded operations in the form (2) are able to approximate a wide range of graph inference tasks. In particular, inspired by message-passing algorithms, Kearnes et al. (2016); Gilmer et al. (2017) generalized the GNN to also learn edge features $\\tilde { A } ^ { ( k ) }$ from the current node hidden representation: ",
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"text": "$$\n\\tilde { A } _ { i , j } ^ { ( k ) } = \\varphi _ { \\tilde { \\theta } } ( \\mathbf { x } _ { i } ^ { ( k ) } , \\mathbf { x } _ { j } ^ { ( k ) } ) ,\n$$",
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"image_caption": [
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"Figure 2: Graph Neural Network ilustration. The Adjacency matrix is computed before every Convolutional Layer. "
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"text": "where $\\varphi$ is a symmetric function parametrized with e.g. a neural network. In this work, we consider a Multilayer Perceptron stacked after the absolute difference between two vector nodes. See eq. 4: ",
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"text": "$$\n\\varphi _ { \\tilde { \\theta } } ( \\mathbf { x } _ { i } ^ { ( k ) } , \\mathbf { x } _ { j } ^ { ( k ) } ) = \\mathbf { M L P } _ { \\tilde { \\theta } } ( a b s ( \\mathbf { x } _ { i } ^ { ( k ) } - \\mathbf { x } _ { j } ^ { ( k ) } ) )\n$$",
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"text": "Then $\\varphi$ is a metric, which is learned by doing a non-linear combination of the absolute difference between the individual features of two nodes. Using this architecture the distance property Symmetry $\\varphi _ { \\tilde { \\theta } } ( a , b ) = \\varphi _ { \\tilde { \\theta } } ( b , a )$ is fulfilled by construction and the distance property Identity $\\varphi _ { \\tilde { \\theta } } ( a , a ) = 0$ is easily learned. ",
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"text": "The trainable adjacency is then normalized to a stochastic kernel by using a softmax along each row. The resulting update rules for node features are obtained by adding the edge feature kernel $\\tilde { A } ^ { ( k ) }$ into the generator family $\\mathcal { A } = \\{ \\tilde { A } ^ { ( k ) } , \\mathbf { 1 } \\}$ and applying (2). Adjacency learning is particularly important in applications where the input set is believed to have some geometric structure, but the metric is not known a priori, such as is our case. ",
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"text": "In general graphs, the network depth is chosen to be of the order of the graph diameter, so that all nodes obtain information from the entire graph. In our context, however, since the graph is densely connected, the depth is interpreted simply as giving the model more expressive power. ",
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"text": "Construction of Initial Node Features The input collection $\\tau$ is mapped into node features as follows. For images $x _ { i } \\in \\mathcal T$ with known label $l _ { i }$ , the one-hot encoding of the label is concatenated with the embedding features of the image at the input of the GNN. ",
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"text": "$$\n\\mathbf { x } _ { i } ^ { ( 0 ) } = \\left( \\phi ( { x } _ { i } ) , h ( l _ { i } ) \\right) ,\n$$",
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"text": "where $\\phi$ is a Convolutional neural network and $h ( l ) \\ \\in \\ \\mathbb { R } _ { + } ^ { K }$ is a one-hot encoding of the label. Architectural details for $\\phi$ are detailed in Section 6.1.1 and 6.1.2. For images $\\tilde { x } _ { j } , \\bar { x } _ { j ^ { \\prime } }$ with unknown label $l _ { i }$ , we modify the previous construction to account for full uncertainty about the label variable by replacing $h ( l )$ with the uniform distribution over the $K$ -simplex: $V _ { j } \\doteq ( \\phi ( \\tilde { x } _ { j } ) , K ^ { - 1 } \\mathbf { 1 } _ { K } )$ , and analogously for $\\bar { x }$ . ",
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"text": "4.3 RELATIONSHIP WITH EXISTING MODELS ",
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"text": "The graph neural network formulation of few-shot learning generalizes a number of recent models proposed in the literature. ",
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"text": "Siamese Networks Siamese Networks Koch et al. (2015) can be interpreted as a single layer message-passing iteration of our model, and using the same initial node embedding (5) $\\mathbf { x } _ { i } ^ { ( 0 ) } =$ $( \\phi ( x _ { i } ) , h _ { i } )$ , using a non-trainable edge feature ",
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"text": "$$\n\\varphi ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) = \\lVert \\phi ( x _ { i } ) - \\phi ( x _ { j } ) \\rVert , \\tilde { A } ^ { ( 0 ) } = \\mathrm { s o f t m a x } ( - \\varphi ) ,\n$$",
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"text": "and resulting label estimation ",
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"text": "$$\n\\hat { Y } _ { * } = \\sum _ { j } \\tilde { A } _ { * , j } ^ { ( 0 ) } \\langle \\mathbf { x } _ { j } ^ { ( 0 ) } , u \\rangle ,\n$$",
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"text": "with $u$ selecting the label field from $\\mathbf { x }$ . In this model, the learning is reduced to learning image embeddings $\\phi ( x _ { i } )$ whose euclidean metric is consistent with the label similarities. ",
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"text": "Prototypical Networks Prototypical networks Snell et al. (2017) evolve Siamese networks by aggregating information within each cluster determined by nodes with the same label. This operation can also be accomplished with a gnn as follows. we consider ",
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"text": "$$\n\\tilde { A } _ { i , j } ^ { ( 0 ) } = \\left\\{ \\begin{array} { c c } { q ^ { - 1 } } & { \\mathrm { i f } l _ { i } = l _ { j } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.\n$$",
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| 728 |
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|
| 729 |
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|
| 730 |
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|
| 731 |
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"type": "text",
|
| 732 |
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"text": "where $q$ is the number of examples per class, and ",
|
| 733 |
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"bbox": [
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| 734 |
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|
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|
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"type": "equation",
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"img_path": "images/1749c665207ae427297a74a5058fad81c6fb233bc84b87bbf9a15f13b11663a5.jpg",
|
| 744 |
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"text": "$$\n\\mathbf { x } _ { i } ^ { ( 1 ) } = \\sum _ { j } \\tilde { A } _ { i , j } ^ { ( 0 ) } \\mathbf { x } _ { j } ^ { ( 0 ) } ,\n$$",
|
| 745 |
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|
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| 756 |
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"text": "where $\\mathbf { x } ^ { ( 0 ) }$ is defined as in the Siamese Networks. We finally apply the previous kernel $\\tilde { A } ^ { ( 1 ) } =$ softmax $\\left( \\varphi \\right)$ applied to $\\mathbf { x } ^ { ( 1 ) }$ to yield class prototypes: ",
|
| 757 |
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"img_path": "images/02c12048be4d1cd58794d6d6ea2e8f782217110e95ea6dd7e4bb1bc95385ad18.jpg",
|
| 768 |
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"text": "$$\n\\hat { Y } _ { * } = \\sum _ { j } \\tilde { A } _ { * , j } ^ { ( 1 ) } \\langle \\mathbf { x } _ { j } ^ { ( 1 ) } , u \\rangle .\n$$",
|
| 769 |
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"type": "text",
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"text": "Matching Networks Matching networks Vinyals et al. (2016) use a set representation for the ensemble of images in $\\tau$ , similarly as our proposed graph neural network model, but with two important differences. First, the attention mechanism considered in this set representation is akin to the edge feature learning, with the difference that the mechanism attends always to the same node embeddings, as opposed to our stacked adjacency learning, which is closer to Vaswani et al. (2017). In other words, instead of the attention kernel in (3), matching networks consider attention mechanisms of the form A˜(k)∗,j $\\tilde { A } _ { * , j } ^ { ( k ) } = \\varphi ( \\mathbf { x } _ { * } ^ { ( k ) } , \\mathbf { x } _ { j } ^ { ( T ) } )$ , where $\\underset { - } { \\mathbf { x } _ { j } ^ { ( T ) } }$ is the encoding function for the elements of the support set, obtained with bidirectional LSTMs. In that case, the support set encoding is thus computed independently of the target image. Second, the label and image fields are treated separately throughout the model, with a final step that aggregates linearly the labels using a trained kernel. This may prevent the model to leverage complex dependencies between labels and images at intermediate stages. ",
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"type": "text",
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"text": "5 TRAINING ",
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"type": "text",
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"text": "We describe next how to train the parameters of the GNN in the different setups we consider: fewshot learning, semi-supervised learning and active learning. ",
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{
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"type": "text",
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"text": "5.1 FEW-SHOT AND SEMI-SUPERVISED LEARNING ",
|
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"text_level": 1,
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"type": "text",
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"text": "In this setup, the model is asked only to predict the label $Y$ corresponding to the image to classify $\\bar { x } \\in \\tau$ , associated with node $^ *$ in the graph. The final layer of the GNN is thus a softmax mapping the node features to the $K$ -simplex. We then consider the Cross-entropy loss evaluated at node $^ *$ : ",
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"type": "equation",
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"img_path": "images/2e2ce54c630a1a436ab7b51bc8ef3a665fb3bf509061d4d219b7fe5b680c0798.jpg",
|
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"text": "$$\n\\ell ( \\Phi ( \\mathcal { T } ; \\Theta ) , Y ) = - \\sum _ { k } y _ { k } \\log P ( Y _ { * } = y _ { k } \\mid \\mathcal { T } ) .\n$$",
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"bbox": [
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"type": "text",
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"text": "The semi-supervised setting is trained identically — the only difference is that the initial label fields of the node will be filled with the uniform distribution on nodes corresponding to $\\tilde { x } _ { j }$ . ",
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{
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"type": "text",
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| 861 |
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"text": "5.2 ACTIVE LEARNING ",
|
| 862 |
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"text_level": 1,
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| 863 |
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"bbox": [
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|
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"type": "text",
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"text": "In the Active Learning setup, the model has the intrinsic ability to query for one of the labels from $\\{ \\tilde { x } _ { 1 } , \\ldots , \\tilde { x } _ { r } \\}$ . The network will learn to ask for the most informative label in order to classify the sample $\\bar { x } \\in \\mathcal { T }$ . The querying is done after the first layer of the GNN by using a Softmax attention over the unlabeled nodes of the graph. For this we apply a function $g ( \\mathbf { x } _ { i } ^ { ( 1 ) } ) \\in \\mathbb { R } ^ { 1 }$ that maps each unlabeled vector node to a scalar value. Function $g$ is parametrized by a two layers neural network. A Softmax is applied over the $\\{ 1 , \\ldots , r \\}$ scalar values obtained after applying $g$ : ",
|
| 874 |
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|
| 883 |
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"type": "text",
|
| 884 |
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"text": "",
|
| 885 |
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"bbox": [
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| 888 |
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|
| 892 |
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},
|
| 893 |
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{
|
| 894 |
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"type": "equation",
|
| 895 |
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"img_path": "images/40956d2404fba285f4050d0797270ebf4e608119e957200ba8e13de641ac4836.jpg",
|
| 896 |
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"text": "$$\n\\mathrm { A t t e n t i o n } = \\mathrm { S o f t m a x } ( g ( \\mathbf { x } _ { \\{ 1 , \\dots , r \\} } ^ { ( 1 ) } ) )\n$$",
|
| 897 |
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"text_format": "latex",
|
| 898 |
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"bbox": [
|
| 899 |
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| 900 |
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| 901 |
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| 902 |
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],
|
| 904 |
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},
|
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|
| 907 |
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"type": "text",
|
| 908 |
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"text": "In order to query only one sample, we set all elements from the $A t t e n t i o n \\in \\mathbb { R } ^ { r }$ vector to 0 except for one. At test time we keep the maximum value, at train time we randomly sample one value based on its multinomial probability. Then we multiply this sampled attention by the label vectors: ",
|
| 909 |
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"bbox": [
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| 916 |
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},
|
| 917 |
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|
| 918 |
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"type": "equation",
|
| 919 |
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"img_path": "images/28a8495332a2a041232a49fbbff9d86801681cbfb4d03b028c7e9c128b1f5bb5.jpg",
|
| 920 |
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"text": "$$\nw \\cdot h ( l _ { i ^ { * } } ) = \\langle \\mathrm { A t t e n t i o n } ^ { \\prime } , h ( l _ { \\{ 1 , \\dots , r \\} } ) \\rangle\n$$",
|
| 921 |
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"text_format": "latex",
|
| 922 |
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| 931 |
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"type": "text",
|
| 932 |
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"text": "The label of the queried vector $h ( l _ { i ^ { * } } )$ is obtained, scaled by the weight $w \\in ( 0 , 1 )$ . This value is then summed to the current representation $\\mathbf { x } _ { i ^ { * } } ^ { ( 1 ) }$ , since we are using dense connections in our GNN model we can sum this $w \\cdot h ( l _ { i ^ { * } } )$ value directly to where the uniform label distribution was concatenated ",
|
| 933 |
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"bbox": [
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},
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{
|
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"type": "equation",
|
| 943 |
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"img_path": "images/875911c465b5db3c2e3f9b2e38d0b3b1a30909ecc266b76f3a9a1cbeb36f297e.jpg",
|
| 944 |
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"text": "$$\n\\mathbf { x } _ { i ^ { * } } ^ { ( 1 ) } = [ \\mathbf { G c } ( \\mathbf { x } _ { i ^ { * } } ^ { ( 0 ) } ) , \\mathbf { x } _ { i ^ { * } } ^ { ( 0 ) } ] = [ \\mathbf { G c } ( \\mathbf { x } _ { i ^ { * } } ^ { ( 0 ) } ) , ( \\phi ( x _ { i ^ { * } } ) , h ( l _ { i ^ { * } } ) ) ]\n$$",
|
| 945 |
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"text_format": "latex",
|
| 946 |
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"bbox": [
|
| 947 |
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| 950 |
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],
|
| 952 |
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| 953 |
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},
|
| 954 |
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{
|
| 955 |
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"type": "text",
|
| 956 |
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"text": "After the label has been summed to the current node, the information is forward propagated. This attention part is trained end-to-end with the rest of the network by backpropagating the loss from the output of the GNN. ",
|
| 957 |
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"bbox": [
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| 964 |
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},
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| 965 |
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{
|
| 966 |
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"type": "text",
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| 967 |
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"text": "6 EXPERIMENTS ",
|
| 968 |
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"text_level": 1,
|
| 969 |
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| 977 |
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{
|
| 978 |
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"type": "text",
|
| 979 |
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"text": "For the few-shot, semi-supervised and active learning experiments we used the Omniglot dataset presented by Lake et al. (2015) and Mini-Imagenet dataset introduced by Vinyals et al. (2016) which is a small version of ILSVRC-12 Krizhevsky et al. (2012). All experiments are based on the $q$ -shot, $K$ -way setting. For all experiments we used the same values $q$ -shot and $K$ -way for both training and testing. ",
|
| 980 |
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| 987 |
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},
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| 988 |
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|
| 989 |
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"type": "text",
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| 990 |
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"text": "Code available at: https://github.com/vgsatorras/few-shot-gnn ",
|
| 991 |
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"bbox": [
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},
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{
|
| 1000 |
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"type": "text",
|
| 1001 |
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"text": "6.1 DATASETS AND IMPLEMENTATION ",
|
| 1002 |
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"text_level": 1,
|
| 1003 |
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},
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{
|
| 1012 |
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"type": "text",
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| 1013 |
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"text": "6.1.1 OMNIGLOT ",
|
| 1014 |
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"text_level": 1,
|
| 1015 |
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"bbox": [
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|
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| 1022 |
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},
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|
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"type": "text",
|
| 1025 |
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"text": "Dataset: Omniglot is a dataset of 1623 characters from 50 different alphabets, each character/class has been drawn by 20 different people. Following Vinyals et al. (2016) implementation we split the dataset into 1200 classes for training and the remaining 423 for testing. We augmented the dataset by multiples of 90 degrees as proposed by Santoro et al. (2016). ",
|
| 1026 |
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"bbox": [
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| 1033 |
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},
|
| 1034 |
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|
| 1035 |
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"type": "text",
|
| 1036 |
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"text": "Architectures: Inspired by the embedding architecture from Vinyals et al. (2016), following Mishra et al. (2017), a CNN was used as an embedding $\\phi$ function consisting of four stacked blocks of $\\{ 3 \\times 3$ -convolutional layer with 64 filters, batch-normalization, $2 \\times 2$ max-pooling, leaky-relu} the output is passed through a fully connected layer resulting in a 64-dimensional embedding. For the GNN we used 3 blocks each of them composed by 1) a module that computes the adjacency matrix and 2) a graph convolutional layer. A more detailed description of each block can be found at Figure 3. ",
|
| 1037 |
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},
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|
| 1046 |
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"type": "text",
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| 1047 |
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"text": "6.1.2 MINI-IMAGENET ",
|
| 1048 |
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"text_level": 1,
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| 1049 |
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},
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{
|
| 1058 |
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"type": "text",
|
| 1059 |
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"text": "Dataset: Mini-Imagenet is a more challenging dataset for one-shot learning proposed by Vinyals et al. (2016) derived from the original ILSVRC-12 dataset Krizhevsky et al. (2012). It consists of $8 4 \\times 8 4$ RGB images from 100 different classes with 600 samples per class. It was created with the purpose of increasing the complexity for one-shot tasks while keeping the simplicity of a light size dataset, that makes it suitable for fast prototyping. We used the splits proposed by Ravi & Larochelle (2016) of 64 classes for training, 16 for validation and 20 for testing. Using 64 classes for training, and the 16 validation classes only for early stopping and parameter tuning. ",
|
| 1060 |
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},
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| 1068 |
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|
| 1069 |
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"type": "text",
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| 1070 |
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"text": "Architecture: The embedding architecture used for Mini-Imagenet is formed by 4 convolutional layers followed by a fully-connected layer resulting in a 128 dimensional embedding. This light architecture is useful for fast prototyping: ",
|
| 1071 |
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"bbox": [
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{
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"type": "text",
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"text": "$1 \\times \\{ 3 \\times 3$ -conv. layer (64 filters), batch normalization, max pool $( 2 , 2 )$ , leaky relu $\\}$ , $1 \\times \\{ 3 \\times 3$ -conv. layer (96 filters), batch normalization, max $\\mathrm { p o o l } ( 2 , 2 )$ , leaky relu}, $1 \\times \\{ 3 \\times 3$ -conv. layer (128 filters), batch normalization, max $\\mathrm { p o o l } ( 2 , 2 )$ , leaky relu, dropout $\\left( 0 . 5 \\right) \\}$ , $1 \\times \\{ 3 \\times 3$ -conv. layer (256 filters), batch normalization, max $\\mathrm { p o o l } ( 2 , 2 )$ , leaky relu, dropout $\\left( 0 . 5 \\right) \\}$ , $1 \\times \\left\\{ \\begin{array} { r l } \\end{array} \\right.$ fc-layer (128 filters), batch normalization}. ",
|
| 1082 |
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"bbox": [
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"page_idx": 7
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{
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"type": "text",
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"text": "The two dropout layers are useful to avoid overfitting the GNN in Mini-Imagenet dataset. The GNN architecture is similar than for Omniglot, it is formed by 3 blocks, each block is described at Figure 3. ",
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"bbox": [
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"page_idx": 7
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{
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"type": "text",
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"text": "6.2 FEW-SHOT ",
|
| 1104 |
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"text_level": 1,
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"bbox": [
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"type": "text",
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"text": "Few-shot learning experiments for Omniglot and Mini-Imagenet are presented at Table 1 and Table 2 respectively. ",
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"bbox": [
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"text": "We evaluate our model by performing different ${ \\bf q }$ -shot, K-way experiments on both datasets. For every few-shot task $\\tau$ , we sample $K$ random classes from the dataset, and from each class we sample $q$ random samples. An extra sample to classify is chosen from one of that $K$ classes. ",
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"bbox": [
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"type": "text",
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"text": "Omniglot: The GNN method is providing competitive results while still remaining simpler than other methods. State of the art results are reached in the 5-Way and 20-way 1-shot experiments. In the 20-Way 1-shot setting the GNN is providing slightly better results than Munkhdalai & Yu (2017) while still being a more simple approach. The TCML approach from Mishra et al. (2017) is in the same confidence interval for 3 out of 4 experiments, but it is slightly better for the 20-Way 5-shot, although the number of parameters is reduced from ${ \\sim } 5 \\mathbf { M }$ (TCML) to $\\sim 3 0 0 \\mathrm { K }$ (3 layers GNN). ",
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"bbox": [
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"type": "text",
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"text": "At Mini-Imagenet table we are also presenting a baseline ”Our metric learning $\\mathbf { \\chi } + K N N ^ { \\prime }$ where no information has been aggregated among nodes, it is a K-nearest neighbors applied on top of the pair-wise learnable metric $\\bar { \\varphi _ { \\theta } } ( \\mathbf { x } _ { i } ^ { ( 0 ) } , \\mathbf { x } _ { j } ^ { ( 0 ) } )$ and trained end-to-end, this learnable metric is competitive by itself compared to other state of the art methods. Even so, a significant improvement (from $6 4 . 0 2 \\%$ to $6 6 . 4 1 \\%$ ) can be seen for the 5-shot 5-Way Mini-Imagenet setting when aggregating information among nodes by using the full GNN architecture. A variety of embedding functions $\\phi$ are used among the different papers for Mini-Imagenet experiments, in our case we are using a simple network of 4 conv. layers followed by a fully connected layer (Section 6.1.2) which served us to compare between Our GNN and Our metric learning $+ ~ K N N$ and it is useful for fast prototyping. More complex embeddings have proven to produce better results, at Mishra et al. (2017) a deep residual network is used as embedding network $\\phi$ increasing the accuracy considerably. Regarding the TCML architecture in Mini-Imagenet, the number of parameters is reduced from ${ \\sim } 1 1 \\mathbf { M }$ (TCML) to ${ \\sim } 4 0 0 \\mathrm { K }$ (3 layers GNN). ",
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{
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"type": "table",
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"img_path": "images/87222dd54ce9d6c0992f4598f628dae14178d75f83063faf9b2d9855f3039239.jpg",
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"table_caption": [],
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| 1161 |
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"table_footnote": [
|
| 1162 |
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"Table 1: Few-Shot Learning — Omniglot accuracies. Siamese Net results are extracted from Vinyals et al. (2016) reimplementation. "
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| 1163 |
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],
|
| 1164 |
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"table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\"> 5-Way</td><td colspan=\"2\">20-Way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Model Pixels Vinyals et al. (2016)</td><td>41.7%</td><td>63.2%</td><td>26.7%</td><td>42.6%</td></tr><tr><td>Siamese Net Koch et al. (2015)</td><td>97.3%</td><td>98.4%</td><td>88.2%</td><td>97.0%</td></tr><tr><td>Matching Networks Vinyals et al. (2016)</td><td>98.1%</td><td>98.9%</td><td>93.8%</td><td>98.5%</td></tr><tr><td>N.Statistician Edwards & Storkey (2016)</td><td>98.1%</td><td>99.5%</td><td>93.2%</td><td>98.1%</td></tr><tr><td>Res.Pair-Wise Mehrotra & Dukkipati (2017)</td><td>-</td><td>-</td><td>94.8%</td><td>-</td></tr><tr><td>Prototypical Networks Snellet al. (2017)</td><td>97.4%</td><td>99.3%</td><td>95.4%</td><td>98.8%</td></tr><tr><td>ConvNet with Memory Kaiser et al. (2017)</td><td>98.4%</td><td>99.6%</td><td>95.0%</td><td>98.6%</td></tr><tr><td>Agnostic Meta-learner Finn et al. (2017)</td><td>98.7 ±0.4%</td><td>99.9 ±0.3%</td><td>95.8 ±0.3%</td><td>98.9 ±0.2%</td></tr><tr><td>MetaNetworks Munkhdalai& Yu (2017)</td><td>98.9%</td><td>=</td><td>97.0%</td><td></td></tr><tr><td>TCML Mishra et al. (2017)</td><td></td><td>98.96% ±0.20% 99.75% ±0.11% 97.64% ±0.30% 99.36% ±0.18%</td><td></td><td></td></tr><tr><td>Our GNN</td><td>99.2%</td><td>99.7%</td><td>97.4%</td><td>99.0%</td></tr></table>",
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864
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],
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"page_idx": 7
|
| 1172 |
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},
|
| 1173 |
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{
|
| 1174 |
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"type": "table",
|
| 1175 |
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"img_path": "images/cba769c07ae71a49cc6a64b78741abbfb4f2c82cc84c69985d7b84ccd1f297f2.jpg",
|
| 1176 |
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"table_caption": [
|
| 1177 |
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"Table 2: Few-shot learning — Mini-Imagenet average accuracies with $9 5 \\%$ confidence intervals. "
|
| 1178 |
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],
|
| 1179 |
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"table_footnote": [],
|
| 1180 |
+
"table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">5-Way</td></tr><tr><td>1-shot</td><td>5-shot</td></tr><tr><td>Matching Networks Vinyals et al. (2016)</td><td>43.6%</td><td>55.3%</td></tr><tr><td>Prototypical Networks Snel et al. (2017)</td><td>46.61% ±0.78%</td><td>65.77% ±0.70%</td></tr><tr><td>Model Agnostic Meta-learner Finn et al. (2017)</td><td>48.70% ±1.84%</td><td>63.1% ±0.92%</td></tr><tr><td>Meta Networks Munkhdalai & Yu (2017)</td><td>49.21% ±0.96</td><td></td></tr><tr><td>Ravi &Larochelle Ravi & Larochelle (2016)</td><td>43.4% ±0.77%</td><td>60.2% ±0.71%</td></tr><tr><td>TCML Mishra et al. (2017)</td><td>55.71% ±0.99%</td><td>68.88% ±0.92%</td></tr><tr><td>Our metric learning + KNN</td><td>49.44% ±0.28%</td><td>64.02% ±0.51%</td></tr><tr><td>Our GNN</td><td>50.33% ±0.36%</td><td>66.41% ±0.63%</td></tr></table>",
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| 1181 |
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"bbox": [
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},
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{
|
| 1190 |
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"type": "text",
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| 1191 |
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"text": "6.3 SEMI-SUPERVISED ",
|
| 1192 |
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"text_level": 1,
|
| 1193 |
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"bbox": [
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{
|
| 1202 |
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"type": "text",
|
| 1203 |
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"text": "Semi-supervised experiments are performed on the 5-way 5-shot setting. Different results are presented when $20 \\%$ and $40 \\%$ of the samples are labeled. The labeled samples are balanced among classes in all experiments, in other words, all the classes have the same amount of labeled and unlabeled samples. ",
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| 1204 |
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"bbox": [
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},
|
| 1212 |
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{
|
| 1213 |
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"type": "text",
|
| 1214 |
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"text": "Two strategies can be seen at Tables 3 and 4. ”GNN - Trained only with labeled” is equivalent to the supervised few-shot setting, for example, in the 5-Way 5-shot $20 \\%$ -labeled setting, this method is equivalent to the 5-way 1-shot learning setting since it is ignoring the unlabeled samples. ”GNN - Semi supervised” is the actual semi-supervised method, for example, in the 5-Way 5-shot $20 \\%$ - labeled setting, the GNN receives as input 1 labeled sample per class and 4 unlabeled samples per class. ",
|
| 1215 |
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"bbox": [
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|
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"page_idx": 8
|
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},
|
| 1223 |
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{
|
| 1224 |
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"type": "text",
|
| 1225 |
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"text": "Omniglot results are presented at Table 3, for this scenario we observe that the accuracy improvement is similar when adding images than when adding labels. The GNN is able to extract information from the input distribution of unlabeled samples such that only using $20 \\%$ of the labels in a 5-shot semi-supervised environment we get same results as in the $40 \\%$ supervised setting. ",
|
| 1226 |
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|
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"page_idx": 8
|
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|
| 1234 |
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{
|
| 1235 |
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"type": "text",
|
| 1236 |
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"text": "In Mini-Imagenet experiments, Table 4, we also notice an improvement when using semi-supervised data although it is not as significant as in Omniglot. The distribution of Mini-Imagenet images is more complex than for Omniglot. In spite of it, the GNN manages to improve by ${ \\sim } 2 \\%$ in the $20 \\%$ and $40 \\%$ settings. ",
|
| 1237 |
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"page_idx": 8
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},
|
| 1245 |
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{
|
| 1246 |
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"type": "table",
|
| 1247 |
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"img_path": "images/7cb9005803c1ecd4a34de02934cf10c727d4ec05c336c8efed5d6deafd8aa405.jpg",
|
| 1248 |
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"table_caption": [
|
| 1249 |
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"Table 3: Semi-Supervised Learning — Omniglot accuracies. "
|
| 1250 |
+
],
|
| 1251 |
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"table_footnote": [],
|
| 1252 |
+
"table_body": "<table><tr><td></td><td colspan=\"3\">5-Way 5-shot</td></tr><tr><td>Model</td><td>20%-labeled</td><td>40%-labeled</td><td>100 %-labeled</td></tr><tr><td>GNN - Trained only with labeled</td><td>99.18%</td><td>99.59%</td><td>99.71%</td></tr><tr><td>GNN -Semi supervised</td><td>99.59%</td><td>99.63%</td><td>99.71%</td></tr></table>",
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| 1253 |
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"page_idx": 8
|
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},
|
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{
|
| 1262 |
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"type": "table",
|
| 1263 |
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"img_path": "images/b223122af0b98fa76252ceec175d22e622d1c92d6afb3fab831ea8c8485f8a63.jpg",
|
| 1264 |
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"table_caption": [
|
| 1265 |
+
"Table 4: Semi-Supervised Learning — Mini-Imagenet average accuracies with $9 5 \\%$ confidence intervals. "
|
| 1266 |
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],
|
| 1267 |
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"table_footnote": [],
|
| 1268 |
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"table_body": "<table><tr><td></td><td colspan=\"3\">5-Way 5-shot</td></tr><tr><td>Model</td><td>20%-labeled</td><td>40%-labeled</td><td>100 %-labeled</td></tr><tr><td>GNN - Trainedonlywithlabeled</td><td>50.33% ±0.36%</td><td>56.91% ±0.42%</td><td>66.41% ±0.63%</td></tr><tr><td>GNN - Semi supervised</td><td>52.45% ±0.88%</td><td>58.76% ±0.86%</td><td>66.41% ±0.63%</td></tr></table>",
|
| 1269 |
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},
|
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{
|
| 1278 |
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"type": "text",
|
| 1279 |
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"text": "6.4 ACTIVE LEARNING ",
|
| 1280 |
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"text_level": 1,
|
| 1281 |
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"bbox": [
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{
|
| 1290 |
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"type": "text",
|
| 1291 |
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"text": "We performed Active Learning experiments on the 5-Way 5-shot set-up when $20 \\%$ of the samples are labeled. In this scenario our network will query for the label of one sample from the unlabeled ones. The results are compared with the Random baseline where the network chooses a random sample to be labeled instead of one that maximally reduces the loss of the classification task $\\tau$ . ",
|
| 1292 |
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"page_idx": 8
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},
|
| 1300 |
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{
|
| 1301 |
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"type": "text",
|
| 1302 |
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"text": "Results are shown at Table 5. The results of the GNN-Random criterion are close to the Semisupervised results for $20 \\%$ -labeled samples from Tables 3 and 4. It means that selecting one random label practically does not improve the accuracy at all. When using the GNN-AL learned criterion, we notice an improvement of $\\sim 3 . 4 \\%$ for Mini-Imagenet, it means that the GNN manages to correctly choose a more informative sample than a random one. In Omniglot the improvement is smaller since the accuracy is almost saturated and the improving margin is less. ",
|
| 1303 |
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"page_idx": 9
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},
|
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{
|
| 1312 |
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"type": "table",
|
| 1313 |
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"img_path": "images/af8f312e6bfe45541ec2d48eb1b22608ab823fc94a600078f07938ed0c9e2fab.jpg",
|
| 1314 |
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"table_caption": [],
|
| 1315 |
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"table_footnote": [],
|
| 1316 |
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"table_body": "<table><tr><td>Method</td><td>5-Way 5-shot20%-labeled</td><td>Method</td><td>5-Way 5-shot 20%-labeled</td></tr><tr><td>GNN - AL</td><td>99.62%</td><td>GNN - AL</td><td>55.99% ±1.35%</td></tr><tr><td>GNN - Random</td><td>99.59%</td><td>GNN - Random</td><td>52.56% ±1.18%</td></tr></table>",
|
| 1317 |
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"bbox": [
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"page_idx": 9
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},
|
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{
|
| 1326 |
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"type": "text",
|
| 1327 |
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"text": "Table 5: Omniglot (left) and Mini-Imagneet (right), average accuracies are shown at both tables, the GNN-AL is the learned criterion that performs Active Learning by selecting the sample that will maximally reduce the loss of the current classification. The GNN - Random is also selecting one sample, but in this case a random one. Mini-Imagenet results are presented with $9 5 \\%$ confidence intervals. ",
|
| 1328 |
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"page_idx": 9
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},
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{
|
| 1337 |
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"type": "text",
|
| 1338 |
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"text": "7 CONCLUSIONS ",
|
| 1339 |
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"text_level": 1,
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| 1340 |
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| 1347 |
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},
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| 1348 |
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{
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"type": "text",
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| 1350 |
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"text": "This paper explored graph neural representations for few-shot, semi-supervised and active learning. From the meta-learning perspective, these tasks become supervised learning problems where the input is given by a collection or set of elements, whose relational structure can be leveraged with neural message passing models. In particular, stacked node and edge features generalize the contextual similarity learning underpinning previous few-shot learning models. ",
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| 1358 |
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},
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| 1359 |
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{
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| 1360 |
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"type": "text",
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| 1361 |
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"text": "The graph formulation is helpful to unify several training setups (few-shot, active, semi-supervised) under the same framework, a necessary step towards the goal of having a single learner which is able to operate simultaneously in different regimes (stream of labels with few examples per class, or stream of examples with few labels). This general goal requires scaling up graph models to millions of nodes, motivating graph hierarchical and coarsening approaches Defferrard et al. (2016). ",
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| 1362 |
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| 1370 |
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"type": "text",
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"text": "Another future direction is to generalize the scope of Active Learning, to include e.g. the ability to ask questions Rothe et al. (2017), or in reinforcement learning setups, where few-shot learning is critical to adapt to non-stationary environments. ",
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| 1373 |
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},
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| 1381 |
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{
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"type": "text",
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| 1383 |
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"text": "ACKNOWLEDGMENTS ",
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"text_level": 1,
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"text": "This work was partly supported by Samsung Electronics (Improving Deep Learning using Latent Structure). ",
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| 1723 |
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|
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|
parse/train/BJj6qGbRW/BJj6qGbRW_middle.json
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parse/train/BJj6qGbRW/BJj6qGbRW_model.json
ADDED
|
The diff for this file is too large to render.
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parse/train/BJxH22EKPS/BJxH22EKPS_layout.pdf
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ADDED
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parse/train/BkgnhTEtDS/BkgnhTEtDS.md
ADDED
|
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|
| 1 |
+
# FEATURE INTERACTION INTERPRETABILITY: A CASE FOR EXPLAINING AD-RECOMMENDATION SYSTEMS VIA NEURAL INTERACTION DETECTION
|
| 2 |
+
|
| 3 |
+
Michael Tsang1, Dehua Cheng2, Hanpeng $\mathbf { L i u } ^ { 1 }$ , Xue Feng2, Eric Zhou2, and Yan Liu1
|
| 4 |
+
|
| 5 |
+
1Department of Computer Science, University of Southern California {tsangm,hanpengl,yanliu.cs}@usc.edu 2Facebook AI {dehuacheng,xfeng,hanningz}@fb.com
|
| 6 |
+
|
| 7 |
+
# ABSTRACT
|
| 8 |
+
|
| 9 |
+
Recommendation is a prevalent application of machine learning that affects many users; therefore, it is important for recommender models to be accurate and interpretable. In this work, we propose a method to both interpret and augment the predictions of black-box recommender systems. In particular, we propose to interpret feature interactions from a source recommender model and explicitly encode these interactions in a target recommender model, where both source and target models are black-boxes. By not assuming the structure of the recommender system, our approach can be used in general settings. In our experiments, we focus on a prominent use of machine learning recommendation: ad-click prediction. We found that our interaction interpretations are both informative and predictive, e.g., significantly outperforming existing recommender models. What’s more, the same approach to interpret interactions can provide new insights into domains even beyond recommendation, such as text and image classification.
|
| 10 |
+
|
| 11 |
+
# 1 INTRODUCTION
|
| 12 |
+
|
| 13 |
+
Despite their impact on users, state-of-the-art recommender systems are becoming increasingly inscrutable. For example, the models that predict if a user will click on an online advertisement are often based on function approximators that contain complex components in order to achieve optimal recommendation accuracy. The complex components come in the form of modules for better learning relationships among features, such as interactions between user and ad features (Cheng et al., 2016; Guo et al., 2017; Wang et al., 2017; Lian et al., 2018; Song et al., 2018). Although efforts have been made to understand the feature relationships, there is still no method that can interpret the feature interactions learned by a generic recommender system, nor is there a strong commercial incentive to do so.
|
| 14 |
+
|
| 15 |
+
In this work, we identify and leverage feature interactions that represent how a recommender system generally behaves. We propose a novel approach, Global Interaction Detection and Encoding for Recommendation (GLIDER), which detects feature interactions that span globally across multiple data-instances from a source recommender model, then explicitly encodes the interactions in a target recommender model, both of which can be black-boxes. GLIDER achieves this by first utilizing our ongoing work on Neural Interaction Detection (NID) (Tsang et al., 2017) with a data-instance perturbation method called LIME (Ribeiro et al., 2016) over a batch of data samples. GLIDER then explicitly encodes the collected global interactions into a target model via sparse feature crossing.
|
| 16 |
+
|
| 17 |
+
In our experiments on ad-click recommendation, we found that the interpretations generated by GLIDER are illuminating, and the detected global interactions can significantly improve the target model’s prediction performance. Because our interaction interpretation method is very general, we also show that the interpretations are informative in other domains: text, image, graph, and dna modeling.
|
| 18 |
+
|
| 19 |
+
|
| 20 |
+
Figure 1: A simplified overview of GLIDER. $\textcircled{1}$ GLIDER utilizes Neural Interaction Detection and LIME together to interpret feature interactions learned by a source black-box model at a data instance, denoted by the large green plus sign. $\textcircled{2}$ GLIDER identifies interactions that consistently appear over multiple data samples, then explicitly encodes these interactions in a target black-box recommender model $f _ { r e c }$ .
|
| 21 |
+
|
| 22 |
+
Our contributions are as follows:
|
| 23 |
+
|
| 24 |
+
1. We propose feature interaction interpretations of general prediction models via interaction detection.
|
| 25 |
+
2. Based on this approach, we propose GLIDER to detect and explicitly encode global feature interactions in black-box recommender systems. This process is a form of automatic feature engineering.
|
| 26 |
+
3. Through experiments, we demonstrate the overall interpretability of detected feature interactions on a variety of domains and show that the interactions can be leveraged to improve recommendation accuracy.
|
| 27 |
+
|
| 28 |
+
# 2 NOTATIONS AND BACKGROUND
|
| 29 |
+
|
| 30 |
+
Notations: Vectors are represented by boldface lowercase letters, such as $\mathbf { x }$ or $\mathbf { z }$ . The $i$ -th entry of a vector $\mathbf { x }$ is denoted by $x _ { i }$ . For a set $s$ , its cardinality is denoted by $| S |$ .
|
| 31 |
+
|
| 32 |
+
Let $d$ be the number of features in a dataset. An interaction, $\mathcal { T }$ , is a subset of feature indices: ${ \mathcal { T } } \subseteq \{ 1 , 2 , \ldots , d \}$ , where $| \mathcal { T } |$ is always $\geq 2$ . A higher-order interaction always has $| \mathcal { T } | \geq 3$ . For a vector $\mathbf { x } \in \mathbb { R } ^ { d }$ , let $\mathbf { x } _ { \mathcal { I } } \in \mathbb { R } ^ { | \mathcal { I } | }$ be restricted to the dimensions of $\mathbf { x }$ specified by $\mathcal { T }$ .
|
| 33 |
+
|
| 34 |
+
Let a black-box model be $f ( \cdot ) : \mathbb { R } ^ { p } \mathbb { R }$ . A black-box recommender model uses tabular feature types, as discussed later in this section. In classification tasks, we assume $f$ is a class logit. $p$ and $d$ may be different depending on feature transformations.
|
| 35 |
+
|
| 36 |
+
Feature Interactions: By definition, a model $f$ learns a statistical (non-additive) feature interaction $\mathcal { T }$ if and only if $f$ cannot be decomposed into a sum of $| \mathcal { T } |$ arbitrary subfunctions $f _ { i }$ , each excluding a corresponding interaction variable (Friedman et al., 2008; Sorokina et al., 2008; Tsang et al., 2017), i.e., $\begin{array} { r } { f ( \mathbf { x } ) \neq \bar { \sum } _ { i \in \mathcal { T } } f _ { i } ( \mathbf { x } _ { \{ 1 , 2 , \ldots , d \} \backslash i } ) } \end{array}$ .
|
| 37 |
+
|
| 38 |
+
For example, a multiplication between two features, $x _ { 1 }$ and $x _ { 2 }$ , is a feature interaction because it cannot be represented as an addition of univariate functions, i.e., $x _ { 1 } x _ { 2 } \neq f _ { 1 } ( x _ { 2 } ) + f _ { 2 } ( x _ { 1 } )$ .
|
| 39 |
+
|
| 40 |
+
Recommendation Systems: A recommender system, $f _ { r e c } ( \cdot )$ , is a model of two feature types: dense numerical features and sparse categorical features. Since the one-hot encoding of categorical feature $x _ { c }$ can be high-dimensional, it is commonly represented in a low-dimensional embedding $\scriptstyle \mathbf { e } _ { c } \ =$ $o n e \mathbf { \mathcal { - } } h o t ( x _ { c } ) \mathbf { V } _ { c }$ via embedding matrix $\mathbf { V } _ { c }$ .
|
| 41 |
+
|
| 42 |
+
# 3 FEATURE INTERACTIONS IN BLACK-BOX MODELS
|
| 43 |
+
|
| 44 |
+
We start by explaining how to obtain a data-instance level (local) interpretation of feature interactions by utilizing interaction detection on feature perturbations.
|
| 45 |
+
|
| 46 |
+
# 3.1 FEATURE PERTURBATION AND INFERENCE
|
| 47 |
+
|
| 48 |
+
Given a data instance $\mathbf { x } \in \mathbb { R } ^ { p }$ , LIME proposed to perturb the data instance by sampling a separate binary representation $\tilde { \mathbf { x } } \in \{ 0 , 1 \} ^ { d }$ of the same data instance. Let $\xi : \{ 0 , 1 \} ^ { d } \overset { \cdot } { } \mathbb { R } ^ { p }$ be the map from the binary representation to the perturbed data instance. Starting from a binary vector of all ones that map to the original features values in the data instance, LIME uniformly samples the number of random features to switch to 0 or the “off” state. In the data instance, “off” could correspond to a 0 embedding vector for categorical features or mean value over a batch for numerical features. It is possible for $d < p$ by grouping features in the data instance to correspond to single binary features in $\tilde { \bf x }$ . An important step is getting black-box predictions of the perturbed data instances to create a dataset with binary inputs and prediction targets: $\mathcal { D } = \{ ( \tilde { \mathbf { x } } _ { i } , y _ { i } ) ~ | ~ y _ { i } = f ( \xi ( \tilde { \mathbf { x } } _ { i } ) ) , \tilde { \mathbf { x } } _ { i } \in \{ 0 , 1 \} ^ { d } \}$ . Though we use LIME’s approach, the next section is agnostic to the instance perturbation method.
|
| 49 |
+
|
| 50 |
+
# 3.2 FEATURE INTERACTION DETECTION
|
| 51 |
+
|
| 52 |
+
Feature interaction detection is concerned with identifying feature interactions in a dataset (Bien et al., 2013; Purushotham et al., 2014; Lou et al., 2013; Friedman et al., 2008). Typically, proper interaction detection requires a pre-processing step to remove correlated features that adversely affect detection performance (Sorokina et al., 2008). As long as features in dataset $\mathcal { D }$ are generated in an uncorrelated fashion, e.g., through random sampling, we can directly use $\mathcal { D }$ to detect feature interactions from black-box model $f$ at data instance x.
|
| 53 |
+
|
| 54 |
+
# 3.2.1 NEURAL INTERACTION DETECTION
|
| 55 |
+
|
| 56 |
+
$f$ can be an arbitrary function and can generate highly nonlinear targets in $\mathcal { D }$ , so we focus on detecting interactions that could have generic forms. In light of this, we leverage our method, Neural Interaction Detection (NID) (Tsang et al., 2017), which accurately and efficiently detects generic non-additive and arbitrary-order statistical feature interactions. NID detects these interactions by training a lasso-regularized multilayer perceptron (MLP) on a dataset, then identifying the features that have high-magnitude weights to common hidden units. NID is efficient by greedily testing the top-interaction candidates of every order at each of $h$ first-layer hidden units, enabling arbitraryorder interaction detection in $O ( h d )$ tests within one MLP.
|
| 57 |
+
|
| 58 |
+
# 3.2.2 GRADIENT-BASED NEURAL INTERACTION DETECTION
|
| 59 |
+
|
| 60 |
+
Besides the non-additive definition of statistical interaction, a gradient definition also exists based on mixed partial derivatives (Friedman et al., 2008), i.e., a function $F ( \cdot )$ exhibits statistical interaction $\mathcal { T }$ among features $z _ { i }$ indexed by $i _ { 1 } , i _ { 2 } , \dotsc , i _ { | \mathcal { T } | } \in \mathcal { T }$ if
|
| 61 |
+
|
| 62 |
+
$$
|
| 63 |
+
E _ { \mathbf { z } } \left[ \frac { \partial ^ { | \mathcal { T } | } F ( \mathbf { z } ) } { \partial z _ { i _ { 1 } } \partial z _ { i _ { 2 } } \dots \partial z _ { i _ { | \mathcal { T } | } } } \right] ^ { 2 } > 0 .
|
| 64 |
+
$$
|
| 65 |
+
|
| 66 |
+
The advantage of this definition is that it allows exact interaction detection from model gradients (Ai & Norton, 2003); however, this definition contains a computationally expensive expectation, and typical neural networks with ReLU activation functions do not permit mixed partial derivatives. For the task of local interpretation, we only examine a single data instance $\mathbf { x }$ , which avoids the expectation. We turn $F$ into an MLP $g ( \cdot )$ with smooth, infinitely-differentiable activation functions such as softplus, which closely follows ReLU (Glorot et al., 2011). We then train the MLP with the same purpose as $\ S 3 . 2 . 1$ to faithfully capture interactions in perturbation dataset $\mathcal { D }$ . Given these conditions, we define an alternate gradient-based neural interaction detector (GradientNID) as:
|
| 67 |
+
|
| 68 |
+
$$
|
| 69 |
+
\omega ( \mathcal { T } ) = \left( \frac { \partial ^ { | \mathcal { T } | } g ( \tilde { \mathbf { x } } ) } { \partial \tilde { x } _ { i _ { 1 } } \partial \tilde { x } _ { i _ { 2 } } \hdots \partial \tilde { x } _ { i _ { | \mathcal { T } | } } } \right) ^ { 2 } ,
|
| 70 |
+
$$
|
| 71 |
+
|
| 72 |
+
where $\omega$ is the strength of the interaction $\mathcal { T }$ , $\tilde { \mathbf { x } }$ is the representation of $\mathbf { x }$ , and the MLP $g$ is trained on $\mathcal { D }$ . While GradientNID exactly detects interactions from the explainer MLP, it needs to compute interaction strengths $\omega$ for feature combinations that grow exponentially in number as $| \mathcal { T } |$ increases. We recommend restricting GradientNID to low-order interactions.
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Algorithm 1 Global Interaction Detection in GLIDER
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<table><tr><td>Input: dataset B, recommender model frec Output: G = {(Ii,ci)}: global interactions Ii and their counts ci over the dataset</td></tr><tr><td>1:G ← initialize occurrence dictionary for global interactions</td></tr><tr><td>2:for each data sample x within dataset B do</td></tr><tr><td>3: S ← MADEX(frec, X)</td></tr><tr><td>4: G ← increment the occurrence count of Ij ∈ S, ∀j = 1,2,...,|S|</td></tr><tr><td>5: sort G by most frequently occurring interactions</td></tr><tr><td>6:[optional prune subset interactions in G within a target number of interactions K</td></tr></table>
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# 3.3 SCOPE
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Based on $\ S 3 . 1$ and $\ S 3 . 2$ , we define a function, $\mathtt { M A D E X } ( f , \mathbf { x } )$ , that takes as inputs black-box $f$ and data instance $\mathbf { x }$ , and outputs ${ \cal { S } } = \{ { \cal { T } } _ { i } \} _ { i = 1 } ^ { k }$ , a set of top- $k$ detected feature interactions. MADEX stands for “Model-Agnostic Dependency Explainer”.
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In some cases, it is necessary to identify a $k$ threshold. Because of the importance of speed for local interpretations, we simply use a linear regression with additional multiplicative terms to approximate the gains given by interactions in $s$ , where $k$ starts at 0 and is incremented until the linear model’s predictions stop improving.
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# 4 GLIDER: GLOBAL INTERACTION DETECTION AND ENCODING FOR RECOMMENDATION
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We now discuss the different components of GLIDER: detecting global interactions in $\ S 4 . 1$ , then encoding these interactions in recommender systems in $\ S 4 . 2$ . Recommender systems are interesting because they have pervasive application in real-world systems, and their features are often very sparse. By sparse features, we mean features with many categories, e.g., millions of user IDs. The sparsity makes interaction detection challenging especially when applied directly on raw data because the one-hot encoding of sparse features creates an extremely large space of potential feature combinations (Fan et al., 2015).
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# 4.1 GLOBAL INTERACTION DETECTION
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In this section, we explain the first step of GLIDER. As defined in $\ S 3 . 3$ , MADEX takes as input a blackbox model $f$ and data instance $\mathbf { x }$ . In the context of this section, MADEX inputs a source recommender system $f _ { r e c }$ and data instance $\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \ldots , x _ { p } ]$ . $x _ { i }$ is the $i$ -th feature field and is either a dense or sparse feature. $p$ is both the total number of feature fields and the number of perturbation variables $( p = d )$ ). We define global interaction detection as repeatedly running MADEX over a batch of data instances, then counting the occurrences of the same detected interactions, shown in Algorithm 1. The occurrence counts are not only a useful way to rank global interaction detections, but also a sanity check to rule out the chance that the detected feature combinations are random selections.
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One potential concern with Alg. 1 is that it could be slow depending on the speed of MADEX. In our experiments, the entire process took less than one hour when run in parallel over a batch of 1000 samples with $\sim 4 0$ features on a 32-CPU server with 2 GPUs. This algorithm only needs to be run once to obtain the summary of global interactions.
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# 4.2 TRUNCATED FEATURE CROSSES
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The global interaction $\mathcal { T } _ { i }$ , outputted by Alg. 1, is used to create a synthetic feature $x _ { \mathcal { T } _ { i } }$ for a target recommender system. The synthetic feature $x _ { \mathcal { T } _ { i } }$ is created by explicitly crossing sparse features indexed in $\mathcal { T } _ { i }$ . If interaction $\mathcal { T } _ { i }$ involves dense features, we bucketize the dense features before crossing them. The synthetic feature is sometimes called a cross feature (Wang et al., 2017; Luo et al., 2019) or conjunction feature (Rosales et al., 2012; Chapelle et al., 2015).
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In this context, a cross feature is an $n$ -ary Cartesian product among $n$ sparse features. If we denote $\mathcal { X } _ { 1 } , \mathcal { X } _ { 2 } , \ldots , \mathcal { X } _ { n }$ as the set of IDs for each respective feature $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ , then their cross feature $x _ { \{ 1 , . . . , n \} }$ takes on all possible values in
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$$
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\mathcal { X } _ { 1 } \times \dots \times \mathcal { X } _ { n } = \{ ( x _ { 1 } , \dots , x _ { n } ) ~ | ~ x _ { i } \in \mathcal { X } _ { i } , \forall i = 1 , \dots , n \}
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$$
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Accordingly, the cardinality of this cross feature is $\left| { \mathcal { X } } _ { 1 } \right| \times \cdots \times \left| { \mathcal { X } } _ { n } \right|$ and can be extremely large, yet many combinations of values in the cross feature are likely unseen in the training data. Therefore, we generate a truncated form of the cross feature with only seen combinations of values, $\mathbf { x } _ { \mathcal { T } } ^ { ( j ) }$ , where $j$ is a sample index in the training data, and $\mathbf { x } _ { \mathcal { T } } ^ { ( j ) }$ is represented as a sparse ID in the cross feature $x \tau$ . We further reduce the cardinality by requiring the same cross feature ID to occur more than times in a batch of samples, or set to a default ID otherwise. These truncation steps significantly reduce the embedding sizes of each cross feature while maintaining their representation power. Once cross features $\{ x _ { \mathbb { Z } _ { i } } \bar \} _ { i }$ are included in a target recommender system, it can be trained as per usual.
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# 4.3 MODEL DISTILLATION VS. ENHANCEMENT
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There are dual perspectives of GLIDER: as a method for model distillation or model enhancement. If a strong source model is used to detect global interactions which are then encoded in more resourceconstrained target models, then GLIDER adopts a teacher-student type distillation process. If interaction encoding augments the same model where the interactions were detected from, then GLIDER tries to enhance the model’s ability to represent the interactions.
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# 5 RELATED WORKS
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Interaction Interpretations: A variety of methods exist to detect feature interactions learned in specific models but not black-box models. For example, RuleFit (Friedman et al., 2008), Additive Groves (Sorokina et al., 2008), and Tree-Shap (Lundberg et al., 2018) detect interactions specifically in trees; likewise PaD2 (Gevrey et al., 2006) and NID (Tsang et al., 2017) detect interactions in multilayer perceptrons. Some methods have attempted to interpret feature groups in black-box models, such as Anchors (Ribeiro et al., 2018), Agglomerative Contextual Decomposition (Singh et al., 2019), and Context-Aware methods (Singla et al., 2019); however, these methods were not intended to identify feature interactions.
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Explicit Interaction Representation: There are increasingly methods for explicitly representing interactions in models. Cheng et al. (2016), Guo et al. (2017), Wang et al. (2017), and Lian et al. (2018) directly incorporate multiplicative cross terms in neural network architectures and Song et al. (2018) use attention as an interaction module, all of which are intended to improve the neural network’s function approximation. This line of work found that predictive performance can improve with dedicated interaction modeling. Luo et al. (2019) followed up by proposing feature sets from data then explicitly encoding them via feature crossing, but this method’s proposals are limited by beam search. Our work approaches this problem from a model interpretation standpoint.
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Black-Box Local vs. Global Interpretations: Data-instance level local interpretation methods are more flexible at explaining general black-box models; however, global interpretations, which cover multiple data instances, have become increasingly desirable to summarize model behavior. Locally Interpretable Model-Agnostic Explanations (LIME) (Ribeiro et al., 2016) and Integrated Gradients (Sundararajan et al., 2017) are some of the most used methods to locally interpret any classifier and neural predictor respectively. There are some methods for global black-box interpretations, such as shuffle-based feature importance (Fisher et al., 2018), submodular pick (Ribeiro et al., 2016), and visual concept extraction (Kim et al., 2018). Our work offers a new tooling option.
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# 6 EXPERIMENTS
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# 6.1 SETUP
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In our experiments, we study interaction interpretation and encoding on real-world data. The hyperparameters in MADEX are as follows. For all experiments, our perturbation datasets $\mathcal { D }$ contain 5000 training samples and 500 samples for each validation and testing. Our usage of NID or GradientNID as the interaction detector (§3.2) depends on the experimental setting. For all experiments that only examine single data instances, we use GradientNID for its exactness and pairwise interaction detection; otherwise, we use NID for its higher-order interaction detection. The MLPs for NID and GradientNID have architectures of 256-128-64 first-to-last hidden layer sizes, and they are trained with learning rate of $\mathrm { 1 e - 2 }$ , batchsize of 100, and the ADAM optimizer. NID uses ReLU activations and an $\ell _ { 1 }$ regularization of $\lambda _ { 1 } = 1 \mathrm { e } { - 4 }$ , whereas GradientNID uses softplus activations and a structural regularizer as MLP+linear regression, which we found offers strong test performance. In general, models are trained with early stopping on validation sets.
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For LIME perturbations, we need to establish what a binary 0 maps to via $\xi$ in the raw data instance (§3.1). In domains involving embeddings, i.e., sparse features and word embeddings, the 0 (“off”) state is the zeroed embedding vector. For dense features, it is the mean feature value over a batch; for images, the mean superpixel RGB of the image. For our DNA experiment, we use a random nucleotide other than the original one. These settings correspond to what is used in literature (Ribeiro et al., 2016; 2018). In our graph experiment, the nodes within the neighborhood of a test node are perturbed, where each node is zeroed during perturbation.
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# 6.2 EXPERIMENTS ON CTR RECOMMENDATION
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In this section, we provide experiments with GLIDER on models trained for clickthrough-rate (CTR) prediction. The recommender models we study include commonly reported baselines, which all use neural networks: Wide&Deep (Cheng et al., 2016), DeepFM (Guo et al., 2017), Deep&Cross (Wang et al., 2017), xDeepFM (Lian et al., 2018), and AutoInt (Song et al., 2018).
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Table 1: CTR dataset statistics
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<table><tr><td>Dataset</td><td>#Samples</td><td>#Features</td><td>Total # Sparse IDs</td></tr><tr><td>Criteo</td><td>45,840,617</td><td>39</td><td>998,960</td></tr><tr><td>Avazu</td><td>40,428,967</td><td>23</td><td>1,544,428</td></tr></table>
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AutoInt is the reported state-of-the-art in academic literature, so we use the model settings and data splits provided by AutoInt’s official public repository1. For all other recommender models, we use public implementations2 with the same original architectures reported in literature, set all embedding sizes to 16, and tune the learning rate and optimizer to reach or surpass the test logloss reported by the AutoInt paper (on AutoInt’s data splits). From tuning, we use the Adagrad optimizer (Duchi et al., 2011) with learning rate of 0.01. All models use early stopping on validation sets.
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The datasets we use are benchmark CTR datasets with the largest number of features: Criteo3 and Avazu4, whose data statistics are shown in Table 1. Criteo and Avazu both contain $4 0 +$ millions of user records on clicking ads, with Criteo being the primary benchmark in CTR research (Cheng et al., 2016; Guo et al., 2017; Wang et al., 2017; Lian et al., 2018; Song et al., 2018; Luo et al., 2019).
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# 6.2.1 GLOBAL INTERACTION DETECTION
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Figure 2: Occurrence counts (Total: 1000) vs. rank of detected interactions from AutoInt on Criteo and Avazu datasets. \* indicates a higher-order interaction (details in
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For each dataset, we train a source AutoInt model, $f _ { r e c }$ , then run global interaction detection via Algorithm 1 on a batch of 1000 samples from the validation set. A full global detection experiment finishes in less than one hour when run in parallel on either Criteo or Avazu datasets in a 32-CPU Intel Xeon E5-2640 v2 $\textcircled { a } \ 2 . 0 0 \mathrm { G H z }$ server with 2 Nvidia 1080 Ti GPUs. The detection results across datasets are shown in Figure 2 Appendix G). as plots of detection counts versus rank. Because the Avazu dataset contains non-anonymized features, we directly show its top-10 detected global interactions in Table 2a.
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Table 2: Understanding feature interactions: top global feature interactions for (a) an ad targeting system via Algorithm 1 and (b) a text sentiment analyzer via $\ S 6 . 3 . 2$ (later). The tables are juxtaposed to assist in understanding feature interactions, i.e., nuanced changes among interacting variables lead to significant changes in prediction probabilities. The prediction outcomes are ad-clicks by users for (a) and text sentiment for (b).
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(a) Explanation of an ad targeting system
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<table><tr><td>Count (Total:1000)</td><td>Interaction</td></tr><tr><td>525</td><td>{device_ip,hour}</td></tr><tr><td>235</td><td>{device_id,device_ip,hour}</td></tr><tr><td>217</td><td>{device_id,app-id}</td></tr><tr><td>203</td><td>{device_ip,device_model, hour}</td></tr><tr><td>194</td><td>{site_id, site_domain}</td></tr><tr><td>190</td><td>{site_id, hour}</td></tr><tr><td>187</td><td>{device_ip,site_id,hour}</td></tr><tr><td>183</td><td>{site_id,site_domain,hour}</td></tr><tr><td>179</td><td>{device_id,hour}</td></tr><tr><td>179</td><td>{device_id,device_ip,device_model,hour}</td></tr></table>
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(b) Explanation of a sentiment analyzer
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<table><tr><td>Count (Total:40)</td><td>Interaction (ordered)</td></tr><tr><td>36</td><td>never, fails</td></tr><tr><td>30</td><td>suspend,disbelief</td></tr><tr><td>30</td><td>too, bad</td></tr><tr><td>29</td><td>very, funny</td></tr><tr><td>29</td><td>neither, nor</td></tr><tr><td>28</td><td>not, miss</td></tr><tr><td>27</td><td>recent, memory</td></tr><tr><td>27</td><td>not, good</td></tr><tr><td>26</td><td>no,denying</td></tr><tr><td>25</td><td>not, bad</td></tr></table>
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From Figure 2, we see that the same interactions are detected very frequently across data instances, and many of the interactions are higher-order interactions. The interaction counts are very significant. For example, any top-1 occurrence count $> 2 5$ is significant for the Criteo dataset $( p < 0 . 0 5 ) $ , and likewise $> 7 1$ for the Avazu dataset, assuming a conservative search space of only up to 3-way interactions ( $| \mathcal { I } | \le 3 )$ . Our top-1 occurrence counts are 691 $\left( \gg 2 5 \right)$ ) for Criteo and 525 $( \gg 7 1 )$ ) for Avazu.
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In Table 2a, the top-interactions are explainable. For example, the interaction between “device ip” and “hour” (in UTC time) makes sense because users - here identified by IP addresses - have ad-click behaviors dependent on their time zones. This is a general theme with many of the top-interactions5. As another example, the interaction between “device id” and “app id” makes sense because ads are targeted to users based on the app they’re in.
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# 6.2.2 INTERACTION ENCODING
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Based on our results from the previous section $( \ S 6 . 2 . 1 )$ , we turn our attention to explicitly encoding the detected global interactions in target baseline models via truncated feature crosses (detailed in $\ S 4 . 2 )$ . In order to generate valid cross feature IDs, we bucketize dense features into a maximum of 100 bins before crossing them and require that final cross feature IDs occur more than $T = 1 0 0$ times over a training batch of one million samples.
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We take AutoInt’s top- $K$ global interactions on each dataset from $\ S 6 . 2 . 1$ with subset interactions excluded (Algorithm 1, line 6) and encode the interactions in each baseline model including AutoInt itself. $K$ is tuned on valiation sets, and model hyperparameters are the same between a baseline and one with encoded interactions. We set $K = 4 0$ for Criteo and $K = 1 0$ for Avazu.
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In Table 3, we found that GLIDER often obtains significant gains in performance based on standard deviation, and GLIDER often reaches or exceeds a desired 0.001 improvement for the Criteo dataset (Cheng et al., 2016; Guo et al., 2017; Wang et al., 2017; Song et al., 2018). The improvements are especially visible with DeepFM on Criteo. We show how this model’s test performance varies with different $K$ in Figure 3. All performance gains are obtained at limited cost of extra model parameters (Table 4) thanks to the truncations applied to our cross features. To avoid extra parameters entirely, we recommend feature selection on the new and existing features.
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One one hand, the evidence that AutoInt’s detected interactions can improve other baselines’ performance suggests the viability of interaction distillation. On the other hand, evidence that AutoInt’s performance on Criteo can improve using its own detected interactions suggests that AutoInt may benefit from learning interactions more explicitly. In either model distillation or enhancement settings, we found that GLIDER performs especially well on industry production models trained on large private datasets with thousands of features.
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Table 3: Test prediction performance by encoding top- $K$ global interactions in baseline recommender systems on the Criteo and Avazu datasets (5 trials). $K$ are 40 and 10 for Criteo and Avazu respectively. $^ { 6 6 } +$ GLIDER” means the inclusion of detected global interactions to corresponding baselines. The “Setting” column is labeled relative to the source of detected interactions: AutoInt. \* scores by Song et al. (2018).
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<table><tr><td rowspan="2">Setting</td><td rowspan="2">Model</td><td colspan="2">Criteo</td><td colspan="2">Avazu</td></tr><tr><td>AUC</td><td>logloss</td><td>AUC</td><td>logloss</td></tr><tr><td rowspan="7">Distillation</td><td>Wide&Deep +GLIDER</td><td>0.8069 ± 5e-4</td><td>0.4446 ± 4e-4</td><td>0.7794± 3e-4</td><td>0.3804 ± 2e-4</td></tr><tr><td>DeepFM</td><td>0.8080±3e-4 0.8079 ± 3e-4</td><td>0.4436 ± 3e-4 0.4436± 2e-4</td><td>0.7795 ± 1e-4 0.7792 ± 3e-4</td><td>0.3802 ± 9e-5 0.3804±9e-5</td></tr><tr><td>+ GLIDER</td><td>0.8097± 2e-4</td><td>0.4420±2e-4</td><td>0.7795 ± 2e-4</td><td>0.3802 ± 2e-4</td></tr><tr><td>Deep&Cross</td><td>0.8076 ± 2e-4</td><td>0.4438± 2e-4</td><td>0.7791 ± 2e-4</td><td>0.3805± 1e-4</td></tr><tr><td>+ GLIDER</td><td>0.8086±3e-4</td><td>0.4428± 2e-4</td><td>0.7792 ± 2e-4</td><td>0.3803 ± 9e-5</td></tr><tr><td>xDeepFM</td><td>0.8084± 2e-4</td><td>0.4433 ± 2e-4</td><td>0.7785± 3e-4</td><td>0.3808 ± 2e-4</td></tr><tr><td>+ GLIDER</td><td>0.8097±3e-4</td><td>0.4421± 3e-4</td><td>0.7787 ± 4e-4</td><td>0.3806± 1e-4</td></tr><tr><td rowspan="2">Enhancement</td><td>AutoInt *</td><td>0.8083</td><td>0.4434</td><td>0.7774</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>0.3811</td></tr><tr><td></td><td>+ GLIDER</td><td>0.8090±2e-4</td><td>0.4426± 2e-4</td><td>0.7773 ± 1e-4</td><td>0.3811 ± 5e-5</td></tr></table>
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Table 4: # parameters of the models in Table 3. M denotes million.
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<table><tr><td>Model</td><td>Criteo</td><td>Avazu</td></tr><tr><td>Wide&Deep</td><td>18.1M</td><td>27.3M</td></tr><tr><td>+ GLIDER</td><td>19.3M (+6.8%)</td><td>27.6M (+1.0%)</td></tr><tr><td>DeepFM + GLIDER</td><td>17.5M</td><td>26.7M</td></tr><tr><td></td><td>18.3M (+4.8%)</td><td>26.9M(+0.6%)</td></tr><tr><td>Deep&Cross + GLIDER</td><td>17.5M 18.7M (+6.9%)</td><td>26.1M 26.4M (+1.0%)</td></tr><tr><td>xDeepFM</td><td></td><td></td></tr><tr><td>+ GLIDER</td><td>18.5M 21.7M(+17.2%)</td><td>27.6M 28.3M (+2.5%)</td></tr><tr><td></td><td></td><td></td></tr><tr><td>AutoInt</td><td>16.4M</td><td>25.1M</td></tr><tr><td>+ GLIDER</td><td>17.3M (+5.1%)</td><td>25.2M(+0.6%)</td></tr></table>
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Figure 3: Test logloss vs. $K$ of DeepFM on the Criteo dataset (5 trials).
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# 6.3 INTERPRETATIONS ON OTHER DOMAINS
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Since the proposed interaction interpretations are not entirely limited to recommender systems, we demonstrate interpretations on more general black-box models. Specifically, we experiment with the function MADEX $( \cdot )$ defined in $\ S 3 . 3$ , which inputs a black-box $f$ , data-instance x, and outputs a set of top- $k$ interactions. The models we use are trained on very different tasks, i.e., ResNet152: an image classifier pretrained on ImageNet ‘14 (Russakovsky et al., 2015; He et al., 2016), Sentiment-LSTM: a 2-layer bi-directional long short-term memory network (LSTM) trained on the Stanford Sentiment Treebank (SST) (Socher et al., 2013; Tai et al., 2015), DNA-CNN: a 2-layer 1D convolutional neural network (CNN) trained on MYC-DNA binding data6 (Mordelet et al., 2013; Yang et al., 2013; Alipanahi et al., 2015; Zeng et al., 2016; Wang et al., 2018; Barrett et al., 2012), and GCN: a 3-layer Graph Convolutional Network trained on the Cora dataset (Kipf & Welling, 2016; Sen et al., 2008). In order to make informative comparisons to the linear LIME baseline, we use LIME’s sample weighting strategy and kernel size (0.25) in this section. We first provide quantitative validation for the detected interactions of all four models in $\ S 6 . 3 . 1$ , followed by qualitative results for ResNet152, Sentiment-LSTM, and DNA-CNN in $\ S 6 . 3 . 2$ .
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# 6.3.1 QUANTITATIVE
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To quantitatively validate our interaction interpretations of general black-box models, we measure the local explanation fidelity of the interactions via prediction performance. As suggested in $\ S 3 . 3$ and $\ S 4 . 2$ , encoding feature interactions is a way to increase a model’s function representation, but this also means that prediction performance gains over simpler first-order models (e.g., linear regression) is a way to test the significance of the detected interactions. In this section, we use neural network function approximators for each top-interaction from the ranking $\{ \mathcal { T } _ { i } \}$ given by MADEX’s interaction detector (in this case NID). Similar to the $k$ -thresholding description in $\ S 3 . 3$ , we start at $k = 0$ , which is a linear regression, then increment $k$ with added MLPs for each $\mathcal { T } _ { i }$ among $\{ \mathcal { T } _ { i } \} _ { i = 1 } ^ { k }$ until validation performance stops improving, denoted at $k = L$ . The MLPs all have architectures of 64-32-16 first-to-last hidden layer sizes and use the binary perturbation dataset $\mathcal { D }$ (from $\ S 3 . 1 \rrangle$ .
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Table 5: Prediction performance (mean-squared error; lower is better) with $( k > 0 )$ ) and without $( k \ = \ 0$ ) interactions for random data instances in the test sets of respective black-box models. $k = L$ corresponds to the interaction at a rank threshold. $2 \le k < L$ are excluded because not all instances have 2 or more interactions. Only results with detected interactions are shown. At least $9 4 \%$ $\left( \geq 1 8 8 \right)$ of the data instances had interactions across 5 trials for each model and score statistic.
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<table><tr><td></td><td>k</td><td>DNA-CNN</td><td>Sentiment-LSTM</td><td>ResNet152</td><td>GCN</td></tr><tr><td>linear LIME</td><td>0</td><td>10e-3±le-3</td><td>8.0e-2±6e-3</td><td>1.9 ±0.1</td><td>7.1e3±7e2</td></tr><tr><td>MADEX (ours)</td><td>1</td><td>8e-3±2e-3</td><td>3.8e-2±6e-3</td><td>1.7 ± 0.1</td><td>5.7e3± 7e2</td></tr><tr><td>MADEX (ours)</td><td>L</td><td>5.0e-3±8e-4</td><td>0.4e-2±3e-3</td><td>0.9± 0.2</td><td>2e3±1e3</td></tr></table>
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Test prediction performances are shown in Table 5 for $k \in \{ 0 , 1 , L \}$ . The average number of features of $\mathcal { D }$ among the black-box models ranges from 18 to 112. Our quantitative validation shows that adding feature interactions for DNA-CNN, SentimentLSTM, and ResNet152, and adding node interactions for GCN result in significant performance gains when averaged over 40 randomly selected data instances in the test set.
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# 6.3.2 QUALITATIVE
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For our qualitative analysis, we provide interaction interpretations via MADEX $( \cdot )$ of ResNet152, SentimentLSTM, and DNA-CNN on test samples. The interpretations are given by $\stackrel { \cdot } { S } = \{ { \cal T } _ { i } \} _ { i = 1 } ^ { k }$ , a set of $k$ detected interactions, which are shown in Figure 4 for ResNet152 and SentimentLSTM. For reference, we also show the top “main effects” by LIME’s original linear regression, which select the top-5 features that attribute towards the predicted class7.
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In Figure 4a, the “interaction” columns show selected features from MADEX’s interactions between Quickshift superpixels (Vedaldi & Soatto, 2008; Ribeiro et al., 2016). To reduce the number of interactions per image, we merged interactions that have overlap coefficient $\geq ~ 0 . 5$ (Vijaymeena & Kavitha, 2016). From (a) ResNet152 interpretations
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top prediction: trolleybus, trolley coach, trackless trolley
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Figure 4: Qualitative examples (more in Appendix D & E)
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<table><tr><td rowspan="2">Original sentence</td><td rowspan="2">Predi- ction</td><td rowspan="2">Main effects</td><td colspan="2">Interactions (ours)</td></tr><tr><td>I</td><td>I</td></tr><tr><td>It never fails to engage us.</td><td>pos.</td><td>never, us</td><td>never, fails</td><td></td></tr><tr><td>The movie makes absolutely no sense.</td><td>neg.</td><td>no, sense</td><td>absolutely, no</td><td>no, sense</td></tr><tr><td>The central story lacks punch.</td><td>neg.</td><td>lacks</td><td>story, lacks</td><td>lacks, punch</td></tr></table>
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the figure, we see that the interactions form a single region or multiple regions of the image. They also tend to be complementary to LIME’s main effects and are sometimes more informative. For example, the interpretations of the “shark” classification show that interaction detection finds the shark fin whereas main effects do not. Interpretations of Sentiment-LSTM are shown in Figure 4b, excluding common stop words (Appendix C). We again see the value of MADEX’s interactions, which show salient combinations of words, such as “never, fails”, “absolutely, no”, and “lacks, punch”.
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In our experiments on DNA-CNN, we consistently detected the interaction between “CACGTG” nucleotides, which form a canonical DNA sequence (Staiger et al., 1989). The interaction was detected $9 7 . 3 \%$ out of 187 CACGTG appearances in the test set.
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In order to run consistency experiments now on Sentiment-LSTM, word interactions need to be detected consistently across different sentences, which na¨ıvely would require an exorbitant amount of sentences. Instead, we initially collect interaction candidates by running MADEX over all sentences in the SST test set, then select the word interactions that appear multiple times. We assume that word interactions are ordered but not necessarily adjacent or positionally bound, e.g., (not, good) $\ne ( \mathrm { g o o d }$ , not), but their exact positions don’t matter. We use the larger IMDB dataset (Maas et al., 2011) to collect different sets of sentences that contain the same ordered words as each interaction candidate (but the sentences are otherwise random). The ranked detection counts of the target interactions on their individual sets of sentences are shown in Table 2b. The average sentence length is 33 words, and interaction occurrences are separated by 2 words on average.
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# 7 CONCLUSION
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We proposed a way to interpret feature interactions in general prediction models, and we proposed GLIDER to detect and encode these interactions in black-box recommender systems. In our experiments on recommendation, we found that our detected global interactions are explainable and that explicitly encoding them can improve predictions. We further validated our interaction interpretations on image, text, graph, and dna models. We hope the interpretations encourage investigation into the complex behaviors of prediction models, especially models with large societal impact. Some opportunities for future work are generating correct attributions for interaction interpretations, preventing false-positive interactions from out-of-distribution feature perturbations, and performing interaction distillation from multiple models rather than just one.
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# ACKNOWLEDGMENTS
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We would like to sincerely thank everyone who has provided their generous feedback for this work. Thank you Youbang Sun, Dongxu Ren, and Beibei Xin for offering early-stage brainstorming and prolonged discussions. Thank you Yuping Luo for providing advice on theoretical analysis of model interpretation. Thank you Rich Caruana for your support and insight. Thank you Artem Volkhin, Levent Ertoz, Ellie Wen, Long Jin, Dario Garcia, and the rest of the Facebook personalization team for your feedback on the paper content. Last but not least, thank you anonymous reviewers for your thorough comments and suggestions. This work was supported by National Science Foundation Awards IIS-1254206 and IIS-1539608, granted to co-author Yan Liu in her academic role at the University of Southern California.
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# A EFFECT OF EXTRA PARAMETERS BY INTERACTION ENCODINGS VS. ENLARGED EMBEDDINGS
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In this section, we study whether increasing embedding size can obtain similar prediction performance gains as explicitly encoding interactions via GLIDER. We increase the embedding dimension sizes of every sparse feature in baseline recommender models to match the total number of model parameters of baseline $^ +$ GLIDER as close as possible. The embedding sizes we used to obtain similar parameter counts are shown in Table 6. For the Avazu dataset, all of the embedding sizes remain unchanged because they were already the target size. The corresponding prediction performances of all models are shown in Table 7. We observed that directly increasing embedding size / parameter counts generally did not give the same level of performance gains that GLIDER provided.
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Table 6: Comparison of # model parameters between baseline models with enlarged embeddings and original baselines $^ +$ GLIDER (from Tables 3 and 4). The models with enlarged embeddings are denoted by the asterick $( ^ { * } )$ . The embedding dimension of sparse features is denoted by “emb. size”. Percent differences are relative to baseline\* models. M denotes million, and the ditto mark (”) means no change in the above line.
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<table><tr><td rowspan="2">Model</td><td colspan="2">Criteo</td><td colspan="2">Avazu</td></tr><tr><td>emb. size</td><td># params</td><td>emb. size</td><td># params</td></tr><tr><td>Wide&Deep*</td><td>17</td><td>19.1M</td><td>16</td><td>27.3M</td></tr><tr><td>Wide&Deep</td><td>16</td><td>18.1M</td><td>16</td><td>；</td></tr><tr><td>+ GLIDER</td><td>16</td><td>19.3M (+1.1%)</td><td>16</td><td>27.6M (+1.0%)</td></tr><tr><td>DeepFM*</td><td>17</td><td>18.5M</td><td>16</td><td>26.7M</td></tr><tr><td>DeepFM</td><td>16</td><td>17.5M</td><td>16</td><td>；</td></tr><tr><td>+GLIDER</td><td>16</td><td>18.3M(-0.9%)</td><td>16</td><td>26.9M (+0.6%)</td></tr><tr><td>Deep&Cross*</td><td>17</td><td>18.5M</td><td>16</td><td>26.1M</td></tr><tr><td>Deep&Cross</td><td>16</td><td>17.5M</td><td>16</td><td>；</td></tr><tr><td>+ GLIDER</td><td>16</td><td>18.7M(+1.0%)</td><td>16</td><td>26.4M (+1.0%)</td></tr><tr><td>xDeepFM*</td><td>19</td><td>21.5M</td><td>16</td><td>27.6M</td></tr><tr><td>xDeepFM</td><td>16</td><td>18.5M</td><td>16</td><td>；</td></tr><tr><td>+GLIDER</td><td>16</td><td>21.7M (+0.7%)</td><td>16</td><td>28.3M (+2.5%)</td></tr><tr><td>AutoInt*</td><td>17</td><td>17.4M</td><td>16</td><td>25.1M</td></tr><tr><td>AutoInt</td><td>16</td><td>16.4M</td><td>16</td><td>；</td></tr><tr><td>+ GLIDER</td><td>16</td><td>17.3M(-1.0%)</td><td>16</td><td>25.2M (+0.6%)</td></tr></table>
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Table 7: Test prediction performance corresponding to the models shown in Table 6
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<table><tr><td rowspan="2">Model</td><td colspan="2">Criteo</td><td colspan="2">Avazu</td></tr><tr><td>AUC</td><td>logloss</td><td>AUC</td><td>logloss</td></tr><tr><td>Wide&Deep*</td><td>0.8072 ± 3e-4</td><td>0.4443 ± 2e-4</td><td>0.7794±3e-4</td><td>0.3804± 2e-4</td></tr><tr><td>Wide&Deep</td><td>0.8069± 5e-4</td><td>0.4446± 4e-4</td><td>；</td><td>；</td></tr><tr><td>+ GLIDER</td><td>0.8080 ±3e-4</td><td>0.4436 ± 3e-4</td><td>0.7795± 1e-4</td><td>0.3802 ± 9e-5</td></tr><tr><td>DeepFM*</td><td>0.8080±4e-4</td><td>0.4435 ± 4e-4</td><td>0.7792 ± 3e-4</td><td>0.3804 ± 9e-5</td></tr><tr><td>DeepFM</td><td>0.8079±3e-4</td><td>0.4436± 2e-4</td><td>；</td><td>；</td></tr><tr><td>+GLIDER</td><td>0.8097± 2e-4</td><td>0.4420± 2e-4</td><td>0.7795± 2e-4</td><td>0.3802 ± 2e-4</td></tr><tr><td>Deep&Cross*</td><td>0.8081±2e-4</td><td>0.4434±2e-4</td><td>0.7791± 2e-4</td><td>0.3805 ±1e-4</td></tr><tr><td>Deep&Cross + GLIDER</td><td>0.8076±2e-4</td><td>0.4438 ± 2e-4</td><td>；</td><td>；</td></tr><tr><td></td><td>0.8086 ±3e-4</td><td>0.4428 ± 2e-4</td><td>0.7792 ± 2e-4</td><td>0.3803 ± 9e-5</td></tr><tr><td>xDeepFM* xDeepFM</td><td>0.8088±1e-4</td><td>0.4429 ±1e-4</td><td>0.7785±3e-4 ”</td><td>0.3808± 2e-4</td></tr><tr><td>+ GLIDER</td><td>0.8084± 2e-4</td><td>0.4433 ± 2e-4</td><td></td><td>”</td></tr><tr><td>AutoInt*</td><td>0.8097 ± 3e-4</td><td>0.4421± 3e-4</td><td>0.7787± 4e-4</td><td>0.3806 ± 1e-4</td></tr><tr><td>AutoInt</td><td>0.8087±2e-4</td><td>0.4431± 1e-4</td><td>0.7774±1e-4</td><td>0.3811 ± 8e-5</td></tr><tr><td></td><td>0.8083</td><td>0.4434</td><td>”</td><td>；</td></tr><tr><td>+ GLIDER</td><td>0.8090 ± 2e-4</td><td>0.4426± 2e-4</td><td>0.7773±1e-4</td><td>0.3811 ± 5e-5</td></tr></table>
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# B EFFECT OF DENSE FEATURE BUCKETIZATION
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We examine the effect of dense feature bucketization on cross feature parameter efficiency for the Criteo dataset, which contains 13 dense features. Figure 5 shows the effects of varying the number of dense buckets on the embedding sizes of the cross features involving dense features. Both the effects on the average and individual embedding size are shown. 14 out of 40 of the cross features involved a dense feature. Different cross features show different parameter patterns as the number of buckets increases (Figure 5b). One one hand, the parameter count sometimes increases then asymptotes. Our requirement that a valid cross feature ID occurs more than $T$ times (§4.2) restricts the growth in parameters. On the other hand, the parameter count sometimes decreases, which happens when the dense bucket size becomes too small to satisfy the $T$ occurrence restriction. In all cases, the parameter counts are kept limited, which is important for overall parameter efficiency.
|
| 334 |
+
|
| 335 |
+
|
| 336 |
+
Figure 5: The effects of varying the number of buckets on (a) on the average embedding size of cross features involving dense features and (b) the individual embedding sizes of the same cross features.
|
| 337 |
+
|
| 338 |
+
# C STOP WORDS
|
| 339 |
+
|
| 340 |
+
For all qualitative interpretations on text (in $\ S 6 . 3 . 2$ and Appendix D), we preprocessed sentences to remove stop words. We use the same stop words suggested by Manning et al. (2008), i.e., $\{ \mathbf { a } ,$ , an, and, are, as, at, be, by, for, from, has, he, in, is, it, its, of, on, that, the, to, was, were, will, with $\}$ .
|
| 341 |
+
|
| 342 |
+
# D QUALITATIVE RESULTS ON SENTIMENT-LSTM VS. BERT
|
| 343 |
+
|
| 344 |
+
In this section, we compare the word interactions discovered by MADEX on Sentiment-LSTM versus BERT. These models perform with accuracies of $8 7 \%$ and $9 2 \%$ respectively on the SST test set. We use a public pre-trained BERT, i.e., DistilBERT (Sanh et al., 2019), which is available online8. The interaction detector we use is GradientNID $( \ S 3 . 2 . 2 )$ , and sample weighting is disabled for this comparison. The top-2 interactions for each model are shown in Table 8 on random sentences from the SST test set.
|
| 345 |
+
|
| 346 |
+
Table 8: Top-ranked word interactions $\mathcal { T } _ { i }$ from Sentiment-LSTM and BERT on randomly selected sentences in the SST test set.
|
| 347 |
+
|
| 348 |
+
<table><tr><td rowspan="2">Original sentence</td><td colspan="2">Sentiment-LSTM</td><td colspan="2">BERT</td></tr><tr><td>I</td><td>I</td><td>I</td><td>I</td></tr><tr><td>An intelligent, earnest, intimate film that drops the ball only when it pauses for blunt exposition to make sure you're getting its metaphysical point.</td><td>intelligent, metaphysical</td><td>metaphysical, point</td><td>intelligent, earnest</td><td>drops, ball</td></tr><tr><td>It's not so much enjoyable to watch as it is enlightening to listen to new sides of a previous reality,and to visit with some of the people who were able to make an impact in the theater world.</td><td>not, enjoyable</td><td>not, so</td><td>not, much</td><td>not, enlightening</td></tr><tr><td>Uneasy mishmash of styles and genres.</td><td>uneasy, mishmash</td><td>mishmash, genres</td><td>uneasy, mishmash</td><td>uneasy, styles</td></tr><tr><td>You're better off staying home and watching the X-Files.</td><td>x, files</td><td>off, x</td><td>better, off</td><td>you, off</td></tr><tr><td>If this is the Danish idea of a good time, prospective tourists might want to consider a different destination-some jolly country embroiled ina bloody civil war,perhaps.</td><td>if, this</td><td>if, good</td><td>if, jolly</td><td> jolly, country</td></tr><tr><td>We can see the wheels turning,and we might resent it sometimes,but this is still a nice little picture,made by bright and friendly souls with a lot of good cheer.</td><td>resent, nice</td><td>we,resent</td><td>nice, good</td><td>nice,made</td></tr><tr><td>One of the greatest family-oriented, fantasy-adventure movies ever.</td><td>family, oriented</td><td>greatest, family</td><td>greatest, family</td><td>adventure, movies</td></tr><tr><td>It's so full of wrong choices that all you can do is shake your head in disbelief- and worry about what classic Oliver Parker intends to mangle next time.</td><td>so,wrong</td><td>full, wrong</td><td>so, wrong</td><td>so, full</td></tr><tr><td>Itsmysteries are transparentlyobvious,and it's too slowly paced to be a thriller.</td><td>mysteries, transparently</td><td>paced, thriller</td><td>too, thriller</td><td>too, paced</td></tr><tr><td>This miserable excuse of a movie runs on empty,believing flatbush machismo will get it through.</td><td>miserable, runs</td><td>excuse,get</td><td>runs,empty</td><td>miserable, runs</td></tr></table>
|
| 349 |
+
|
| 350 |
+
# E ADDITIONAL QUALITATIVE RESULTS FOR RESNET152
|
| 351 |
+
|
| 352 |
+
|
| 353 |
+
Figure 6: Additional qualitative results, following Figure 4a, on random test images in ImageNet. Interactions are denoted by $\mathcal { T } _ { i }$ and are unordered. Overlapping interactions with overlap coefficient $\geq 0 . 5$ are merged to reduce $| \{ \mathcal { T } _ { i } \} |$ per test image.
|
| 354 |
+
|
| 355 |
+
# F DETECTION PERFORMANCE OF MADEX VS. BASELINES
|
| 356 |
+
|
| 357 |
+
We compare the detection performances between MADEX and baselines on identifying feature interactions learned by complex models, i.e., XGBoost (Chen & Guestrin, 2016), Multilayer Perceptron (MLP), and Long Short-Term Memory Network (LSTM) (Hochreiter & Schmidhuber, 1997). The baselines are Tree-Shap: a method to identify interactions in tree-based models like XGBoost (Lundberg et al., 2018), MLP-ACD+: a modified version of ACD (Singh et al., 2019; Murdoch et al., 2018) to search all pairs of features in MLP to find the best interaction candidate, and LSTM-ACD+: the same as MLP-ACD $^ +$ but for LSTMs. All baselines are local interpretation methods. For MADEX, we sample continuous features from a truncated normal distribution $\mathcal { N } ( \mathbf { x } , \sigma ^ { 2 } \mathbf { I } )$ centered at a specified data instance $\mathbf { x }$ and truncated at $\sigma$ . Our MADEX experiments consist of two methods, NID and GradNID (shorthand for GradientNID).
|
| 358 |
+
|
| 359 |
+
Table 9: Data generating functions with interactions
|
| 360 |
+
|
| 361 |
+
<table><tr><td>F1(x)=</td><td>10x1x2+</td></tr><tr><td>F2(x)=</td><td>x102+∑3i 10</td></tr><tr><td>F3(x)=</td><td></td></tr><tr><td>F4(x)=</td><td></td></tr></table>
|
| 362 |
+
|
| 363 |
+
We evaluate interaction detection performance by using synthetic data where ground truth interactions are known (Hooker, 2004; Sorokina et al., 2008). We generate 10e3 samples of synthetic data using functions $F _ { 1 } - F _ { 4 }$ (Table 9) with continuous features uniformly distributed between $- 1$ to 1. Next, we train complex models (XGBoost, MLP, and LSTM) on this data. Lastly, we run MADEX and the baselines on 10 trials of 20 data instances at randomly sampled locations on the synthetic function domain. Between trials, the complex models are trained with different random initialization to test the stability of each interpretation method. Interaction detection performance is computed by the average R-precision (Manning et al., $2 0 0 8 ) ^ { 9 }$ of interaction rankings across the sampled data instances.
|
| 364 |
+
|
| 365 |
+
Results are shown in Table 10. MADEX (NID and GradNID) performs well compared to the baselines. On the tree-based model, MADEX can compete with the tree-specific baseline Tree-Shap, which only detects pairwise interactions. On MLP and LSTM, MADEX performs significantly better than $\mathbf { A C D + }$ . The performance gain is especially large in the LSTM setting. Comparing NID and GradNID, NID tends to perform better in this experiment because it takes its entire sampling region into account whereas GradNID examines a single data instance.
|
| 366 |
+
|
| 367 |
+
Table 10: Detection Performance in R-Precision (higher the better). $\sigma = 0 . 6$ (max: 3.2). “Tree” is XGBoost. \*Does not detect higher-order interactions. $\dagger$ Requires an exhaustive search of all feature combinations.
|
| 368 |
+
|
| 369 |
+
<table><tr><td></td><td colspan="3">Tree</td><td colspan="3">MLP</td><td colspan="3">LSTM</td></tr><tr><td></td><td>Tree-Shap</td><td>NID</td><td>GradNID</td><td>MLP-ACD+</td><td>NID</td><td>GradNID</td><td>LSTM-ACD+</td><td>NID</td><td>GradNID</td></tr><tr><td>F1(x）</td><td>1±0</td><td>1±0</td><td>0.96± 0.04</td><td>0.63±0.08</td><td>1±0</td><td>1±0</td><td>0.3±0.2</td><td>1±0</td><td>1±0</td></tr><tr><td>F（x）</td><td>1±0</td><td>0.3± 0.4</td><td>0.6±0.4</td><td>0.41 ± 0.06</td><td>1±0</td><td>0.95 ± 0.04</td><td>0.01±0.02</td><td>0.99 ±0.02</td><td>0.95±0.04</td></tr><tr><td>F（x）</td><td>1±0</td><td>1±0</td><td>1±0</td><td>0.3±0.2</td><td>1±0</td><td>1±0</td><td>0.05±0.08</td><td>1±0</td><td>1±0</td></tr><tr><td>F4(x)</td><td>*</td><td>1±0</td><td>+</td><td>+</td><td>1±0</td><td>+</td><td>+</td><td>1±0</td><td>+</td></tr></table>
|
| 370 |
+
|
| 371 |
+
# G HIGHER-ORDER INTERACTIONS
|
| 372 |
+
|
| 373 |
+
This section shows how often different orders of higher-order interactions are identified by GLIDER / MADEX. Figure 7 plots the occurrence counts of global interactions detected in AutoInt for the Criteo and Avazu dataset, which correspond to the results in Figure 2. Here we show the occurrence counts of higher-order interactions, where the exact interaction cardinality is annotated besides each data point. 3-way interactions are the most common type, followed by 4-, then 5-way interactions.
|
| 374 |
+
|
| 375 |
+
Figure 8 plots histograms of interaction cardinalities for all interactions detected from ResNet152 and Sentiment-LSTM across 1000 random samples in their test sets. The average number of features are 66 and 18 for ResNet152 and Sentiment-LSTM respectively. Higher-order interactions are common in both models.
|
| 376 |
+
|
| 377 |
+
|
| 378 |
+
Figure 7: Occurrence counts (total: 1000) vs. rank of interactions detected from AutoInt on (a) Criteo and (b) Avazu datasets. Each higher-order interaction is annotated with its interaction cardinality.
|
| 379 |
+
|
| 380 |
+
|
| 381 |
+
Figure 8: Histograms of interaction sizes for interactions detected in (a) ResNet152 and (b) Sentiment-LSTM across 1000 random samples in respective test sets.
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parse/train/ByG8A7cee/ByG8A7cee.md
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| 1 |
+
# REFERENCE-AWARE LANGUAGE MODELS
|
| 2 |
+
|
| 3 |
+
Zichao $\mathbf { Y a n g ^ { 1 * } }$ ∗, Phil Blunsom2,3, Chris Dyer1,2, and Wang Ling2 1Carnegie Mellon University, 2DeepMind, and 3University of Oxford zichaoy@cs.cmu.edu, {pblunsom,cdyer,lingwang}@google.com
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
We propose a general class of language models that treat reference as an explicit stochastic latent variable. This architecture allows models to create mentions of entities and their attributes by accessing external databases (required by, e.g., dialogue generation and recipe generation) and internal state (required by, e.g. language models which are aware of coreference). This facilitates the incorporation of information that can be accessed in predictable locations in databases or discourse context, even when the targets of the reference may be rare words. Experiments on three tasks show our model variants outperform models based on deterministic attention.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Referring expressions (REs) in natural language are noun phrases (proper nouns, common nouns, and pronouns) that identify objects, entities, and events in an environment. REs occur frequently and they play a key role in communicating information efficiently. While REs are common, previous works neglect to model REs explicitly, either treating REs as ordinary words in the model or replacing them with special tokens. Here we propose a language modeling framework that explicitly incorporates reference decisions.
|
| 12 |
+
|
| 13 |
+
In Figure 1 we list examples of REs in the context of the three tasks that we consider in this work. Firstly, reference to a database is crucial in many applications. One example is in task oriented dialogue where access to a database is necessary to answer a user’s query (Young et al., 2013; Li et al., 2016; Vinyals & Le, 2015; Wen et al., 2015; Sordoni et al., 2015; Serban et al., 2016; Bordes & Weston, 2016; Williams & Zweig, 2016; Shang et al., 2015; Wen et al., 2016). Here we consider the domain of restaurant recommendation where a system refers to restaurants (name) and their attributes (address, phone number etc) in its responses. When the system says “the nirala is a nice restaurant”, it refers to the restaurant name the nirala from the database. Secondly, many models need to refer to a list of items (Kiddon et al., 2016; Wen et al., 2015). In the task of recipe generation from a list of ingredients (Kiddon et al., 2016), the generation of the recipe will frequently reference these items. As shown in Figure 1, in the recipe “Blend soy milk and . . . ”, soy milk refers to the ingredient summaries. Finally, we address references within a document (Mikolov et al., 2010; Ji et al., 2015; Wang & Cho, 2015), as the generation of words will ofter refer to previously generated words. For instance the same entity will often be referred to throughout a document. In Figure 1, the entity you refers to I in a previous utterance.
|
| 14 |
+
|
| 15 |
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In this work we develop a language model that has a specific module for generating REs. A series of latent decisions (should I generate a RE? If yes, which entity in the context should I refer to? How should the RE be rendered?) augment a traditional recurrent neural network language model and the two components are combined as a mixture model. Selecting an entity in context is similar to familiar models of attention (Bahdanau et al., 2014), but rather than being a deterministic function that reweights representations of elements in the context, it is treated as a distribution over contextual elements which are stochastically selected and then copied or, if the task warrants it, transformed (e.g., a pronoun rather than a proper name is produced as output). Two variants are possible for updating the RNN state: one that only looks at the generated output form; and a second that looks at values of the latent variables. The former admits trivial unsupervised learning, latent decisions are conditionally independent of each other given observed context, whereas the latter enables more expressive models that can extract information from the entity that is being referred to. In each of the three tasks, we demonstrate our reference aware model’s efficacy in evaluations against models that do not explicitly include a reference operation.
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Figure 1: Reference-aware language models.
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Our contributions are as follows:
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• We propose a general framework to model reference in language and instantiate it in the context of dialogue modeling, recipe generation and coreference based language models. • We build three data sets to test our models. There lack existing data sets that satisfy our need, so we build these data sets ourselves. These data sets are either built on top existing data set (we constructed the table for DSTC2 data set for dialogue evaluation), crawled from websites (we crawled all recipes in www.allrecipes.com) or annotated with NLP tools (we annotate the coreference with Gigaword corpus for our evaluation). • We perform comprehensive evaluation of our models on the three data sets and verify our models perform better than strong baselines.
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# 2 REFERENCE-AWARE LANGUAGE MODELS
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Here we propose a general framework for reference-aware language models.
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We denote each document as a series of tokens $x _ { 1 } , \ldots , x _ { L }$ , where $L$ is the number of tokens in the document. Our goal is to maximize the probabilities $p ( x _ { i } \mid c _ { i } )$ , for each word in the document based on its previous context $c _ { i } = x _ { 1 } , \ldots , x _ { i - 1 }$ . In contrast to traditional neural language models, we introduce a variable at each position $z _ { i }$ , which controls the decision on which source $x _ { i }$ is generated from. The token conditional probably is then obtained by:
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$$
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p ( x _ { i } \mid c _ { i } ) = p ( x _ { i } \mid z _ { i } , c _ { i } ) p ( z _ { i } \mid c _ { i } ) .
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$$
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In dialogue modeling and recipe generation, $z _ { i }$ will simply taken on values in $\{ 0 , 1 \}$ . Where $z _ { i } = 1$ denotes that $x _ { i }$ is generated as a reference, either to a database entry or an item in a list. However, $z _ { i }$ can also be defined as a distribution over previous entities, allowing the model to predict $x _ { i }$ conditioned on its a previous mention word. This will be the focus of the coreference language model. When $z _ { i }$ is not observed (which it generally will not be), we will train our model to maximize the marginal probability in Eq. 1 directly.
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# 2.1 DIALOGUE MODEL WITH DATABASE SUPPORT
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We first apply our model on task-oriented dialogue systems in the domain of restaurant recommendations, and work on the data set from the second Dialogue State Tracking Challenge (DSTC2) (Henderson et al., 2014). Table. 1 is one example dialogue from this dataset.
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We can observe from this example, users get recommendations of restaurants based on queries that specify the area, price and food type of the restaurant. We can support the system’s decisions by incorporating a mechanism that allows the model to query the database allowing the model to find restaurants that satisfy the users queries. Thus, we crawled TripAdvisor for restaurants in the
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M: Hello , welcome to the Cambridge restaurant system? You can ask for restaurants by area, price range or food type . How may I help you? U: i want a moderately priced restaurant in the north town M: the nirala is a nice restaurant in the north of town in the moderate price range U: what is the address M: Sure , the nirala is on 7 Milton Road Chesterton
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Table 1: Example dialogue, M stands for Machine and U stands for User
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Table 2: Fragment of database for dialogue system.
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<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICE RANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>ali baba</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>lebanese</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>59 Hills Road City Cen-tre</td><td rowspan=1 colspan=1>CB 2,1NT</td><td rowspan=1 colspan=1>01462 432565</td></tr><tr><td rowspan=1 colspan=1>thenirala</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>north</td><td rowspan=1 colspan=1>7 Milton Road Chester-ton</td><td rowspan=1 colspan=1>CB 4,1 UY</td><td rowspan=1 colspan=1>01223 360966</td></tr></table>
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Cambridge area, where the dialog dataset was collected. Then, we remove restaurants that do not appear in the data set and create a database with 109 entries with restaurants and their attributes (e.g. food type). A sample of our database is shown in Table. 2. We can observe that each restaurant contains 6 attributes that are generally referred in the dialogue dataset. As such, if the user requests a restaurant that serves “indian” food, we wish to train a model that can search for entries whose “food” column contains “indian”. Now, we describe how we deploy a model that fulfills these requirements.
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# 2.1.1 DIALOGUE MODEL
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Figure 2: Hierarchical RNN Seq2Seq model
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We build a model based on the hierarchical RNN model described in (Serban et al., 2016), as in dialogues, the generation of the response is not only dependent on the previous sentence, but on all sentences leading to the response. We assume that a dialogue is alternated between a machine and a user. An illustration of the model is shown in Figure 2.
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Consider a dialogue with $T$ turns, and the utterance from a user is denoted as $X = \{ x _ { i } \} _ { i = 1 } ^ { T }$ , where $i$ is the $i$ -th utterance, whereas the utterance from a machine is denoted as ${ \cal Y } = \{ y _ { i } \} _ { i = 1 } ^ { T }$ , where $i$ is the $i$ -th utterance. We define $x _ { i } = \{ x _ { i j } \} _ { j = 1 } ^ { | x _ { i } | }$ , $y _ { i } = \{ y _ { i v } \} _ { v = 1 } ^ { | y _ { i } | }$ , where $x _ { i j }$ denotes the $j$ -th token in the $i$ -th utterance from the user, whereas $y _ { i v }$ denotes the $v$ -th token in the $i$ -th utterance from the machine. Finally, $\left| x _ { i } \right|$ and $| y _ { i } |$ denote the number of tokens in the user and machine utterances, respectively. The dialogue sequence starts with machine utterance $\left\{ y _ { 1 } , x _ { 1 } , y _ { 2 } , x _ { 2 } , \dots , y _ { T } , x _ { T } \right\}$ . We would like to model the utterances from the machine
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$$
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p ( y _ { 1 } , y _ { 2 } , \ldots , y _ { T } | x _ { 1 } , x _ { 2 } , \ldots , x _ { T } ) = \prod _ { i } p ( y _ { i } | y _ { < i } , x _ { < i } ) = \prod _ { i , v } p ( y _ { i , v } | y _ { i , < v } , y _ { < i } , x _ { < i } ) ,
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$$
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where $y _ { < i }$ denotes all the utterances before $i$ and $y _ { i , < v }$ denotes the first $v - 1$ tokens in the $i$ -th utterance of the machine. A neural model is employed to predict $p ( y _ { i , v } | y _ { i , < v } , y _ { < i } , x _ { < i } )$ , which operates as follows:
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Sentence Encoder: We first encode previous utterances $y _ { < i }$ and $x _ { < i }$ into continuous space by generatistate mploying a LSTM encodand apply the recursion $x _ { i }$ , and sta, where he initial LSTMdenotes a word $h _ { i , 0 } ^ { x }$ $h _ { i , j } ^ { x } = \mathrm { L S T M } _ { \mathrm { E } } ^ { - } ( W _ { E } x _ { i , j } , h _ { i , j - 1 } ^ { x } )$ $W _ { E } x _ { i , j }$
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embedding lookup for the token $x _ { i , j }$ , and $\mathbf { L S T M _ { E } }$ denotes the LSTM transition function described in Hochreiter & Schmidhuber (1997). The representation of the user utterance is represented by the final LSTM state $h _ { i } ^ { x } \ = \ h _ { i , | x _ { i } | } ^ { x }$ The same process is applied to obtain the machine utterance representation $h _ { i } ^ { y } = h _ { i , | y _ { i } | } ^ { y }$ hyi,|yi|
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Turn Encoder: Then, combine all the representations of all the utterances with a second LSTM, which encodes the sequence $\{ h _ { 1 } ^ { y } , h _ { 1 } ^ { x } , . . . , h _ { i } ^ { \bar { y } } , h _ { i } ^ { x } \}$ into a continuous vector. Once again, we start with an initial state $u _ { 0 }$ and feed each of the utterance representation to obtain the following LSTM state, until the final state is obtained. For simplicity, we shall refer to this as $u _ { i }$ , which can be seen as the hierarchical encoding of the previous $i$ utterances.
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Seq2Seq Decoder: As for decoding, in order to generate each utterance $y _ { i }$ , we can feed $u _ { i - 1 }$ into the decoder LSTM as the initial state $s _ { i , 0 } = u _ { i - 1 }$ and decode each token in $y _ { i }$ . Thus, we can express the decoder as:
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$$
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\begin{array} { r l } & { s _ { i , v } ^ { y } = \mathrm { L S T M } _ { \mathrm { D } } ( W _ { E } y _ { i , v - 1 } , s _ { i , v - 1 } ) , } \\ & { p _ { i , v } ^ { y } = \mathrm { s o f t m a x } ( W s _ { i , v } ^ { y } ) , } \end{array}
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$$
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where the desired probability $p ( y _ { i , v } | y _ { i , < v } , y _ { < i } , x _ { < i } )$ is expressed by $p _ { i , v } ^ { y }$
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Attention based decoder: We can also incorporate the attention mechanism in our hierarchical model. An attention model builds a representation $d$ by averaging over a set of vectors $p$ . We define the attention function as $a = \mathrm { A T T N } ( p , q )$ , where $a$ is a probability distribution over the set of vectors $p$ , conditioned on any input representation $q$ . A full description of this operation is described in (Bahon the current decoder state danau et al., 2014). Thus, for each generated token $s _ { i , v } ^ { y }$ , obtaining the attentions over input tokens from previous turn $y _ { i , v }$ , we compute the attentions $a _ { i , v }$ , conditioned $( i - 1 )$ . We denote the vector of all tokens in previous turn as $h _ { i - 1 } ^ { x , y } = [ \{ h _ { i - 1 , j } ^ { x } \} _ { j = 1 } ^ { | x _ { i - 1 } | } , \{ h _ { i - 1 , v } ^ { y } \} _ { v = 1 } ^ { | y _ { i - 1 } | } ]$ }|yi−1|] Let . $K = | h _ { i - 1 } ^ { x , y } |$ be the number of tokens in previous turn. Thus, we obtain the attention probabilities − over all previous tokens $a _ { i , v }$ as $\mathrm { A T T N } ( s _ { i , v } ^ { y } , h _ { i - 1 } ^ { x , y } )$ . Then, the weighted sum is computed over these probabilities $\begin{array} { r } { d _ { i , v } = \sum _ { k \in K } a _ { i , v , k } h _ { i - 1 , k } ^ { x , y } } \end{array}$ , where $_ { a _ { i , v , k } }$ is the probability of aligning to the $k$ -th token from previous turn. The resulting vector $d _ { i , v }$ is used to obtain the probability of the following word $p _ { i , v } ^ { y }$ . Thus, we express the decoder as:
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$$
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\begin{array} { r l } & { s _ { i , v } ^ { y } = \mathrm { L S T M } _ { \mathrm { D } } ( [ { W } _ { \mathrm { E } } y _ { i , v - 1 } , d _ { i , v - 1 } ] , s _ { i , v - 1 } ) , } \\ & { a _ { i , v } = \mathrm { A T T N } ( h _ { i - 1 } ^ { x , y } , s _ { i , v } ^ { y } ) , } \\ & { d _ { i , v } = \displaystyle \sum _ { k \in K } a _ { i , v , k } h _ { i - 1 , k } ^ { x , y } , } \\ & { p _ { i , v } ^ { y } = \mathrm { s o f t m a x } ( W [ s _ { i , v } ^ { y } , d _ { i , v } ] ) . } \end{array}
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$$
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# 2.1.2 INCORPORATING TABLE ATTENTION
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Figure 3: Table based decoder.
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We now extend the attention model in order to allow the attention to be computed over a table, allowing the model to condition the generation on a database.
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We denote a table with $R$ rows and $C$ columns as $\{ f _ { r , c } \} , r \in [ 1 , R ] , c \in [ 1 , C ]$ , where $f _ { r , c }$ is the cell in row $r$ and column $c$ . The attribute of each column is denoted as $s _ { c }$ , where $c$ is the $c$ -th attribute. $f _ { r , c }$ and $s _ { c }$ are one-hot vector.
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Table Encoding: To encode the table, we build an attribute vector $g _ { c }$ for each column. For each cell $f _ { r , c }$ of the table, we concatenate it with the corresponding attribute $g _ { c }$ and then feed it through a one-layer MLP as follows: $g _ { c } = W _ { E } s _ { c }$ and then $e _ { r , c } = \operatorname { t a n h } ( W [ W _ { E } f _ { r , c } , g _ { c } ] )$ .
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Table Attention: The diagram for table attention is shown in Figure 3a. The attention over cells in the table is conditioned on a given vector $q$ , similarly to the attention model for sequences $\mathbf { A T T N } ( p , q )$ . However, rather than a sequence $p$ , we now operate over a table $f$ . Our attention model computes a attribute attention followed by row attention of the table. We first use the attention mechanism on the attributes to find out which attribute the user asks about. Suppose a user says cheap, then we should focus on the price attribute. After we get the attention probability $p ^ { a } = \mathrm { A T T N } ( \{ g _ { c } \} , q )$ , over the attribute, we calculate the weighted representation for each row $\begin{array} { r } { e _ { r } = \sum _ { c } p _ { c } ^ { a } e _ { r c } } \end{array}$ conditioned on $p ^ { a }$ . Then $e _ { r }$ has the price information of each row. We further use attention mechanism on $e _ { r }$ and get the probability $p ^ { r } \bar { \mathbf { \Psi } } = \mathrm { A T T N } ( \{ e _ { r } \} , q )$ over the rows. Then restaurants with cheap price will be picked. Then, using the probabilities $p ^ { r }$ , we compute the weighted average over the all rows $\begin{array} { r } { e _ { c } = \sum _ { r } p _ { r } ^ { r } e _ { r , c } } \end{array}$ , which is used in the decoder. The detailed process is:
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$$
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\begin{array} { r l r } { { p _ { a } = \mathrm { A T T N } ( \{ g _ { c } \} , q ) , } } \\ & { } & { \quad e _ { r } = \displaystyle \sum _ { c } p _ { c } ^ { a } e _ { r c } \quad \forall r , } \\ & { } & { \quad p _ { r } = \mathrm { A T T N } ( \{ e _ { r } \} , q ) , } \\ & { } & { \quad e _ { c } = \displaystyle \sum _ { r } p _ { r } ^ { r } e _ { r , c } \quad \forall c . } \end{array}
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$$
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This is embedded in the decoder by replacing the conditioned state as the current decoder state $s _ { i , 0 } ^ { y }$ and then at each step, conditioning the prediction of ach step. The detailed diagram of table attention is s $y _ { i , v }$ on n in $\{ e _ { c } \}$ by using attention mechanismre 3a.
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# 2.1.3 INCORPORATING TABLE POINTER NETWORKS
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We now describe the mechanism used to refer to specific database entries during decoding. At each timestep, the model needs to decide whether to generate the next token from an entry of the database or from the word softmax. This is performed as follows.
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Pointer Switch: We use $z _ { i , v } \in [ 0 , 1 ]$ to denote the decision of whether to copy one cell from the table. We compute this probability as follows:
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$$
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p ( z _ { i , v } | s _ { i , v } ) = \mathrm { s i g m o i d } ( W [ s _ { i , v } , d _ { i , v } ] ) .
|
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$$
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Thus, if $z _ { i , v } = 1$ , the next token $y _ { i , v }$ will be generated from the database, whereas if $z _ { i , v } = 0$ , then the following token is generated from a softmax. We shall now describe how we generate tokens from the database.
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Table Pointer: If $z _ { i , v } = 1$ , the token is generated from the table. The detailed process of calculating the probability distribution over the table is shown in Figure 3b. This is similar to the attention mechanism, except that we perform a column attention to compute the probabilities of copying from each column after Equation. 5. More formally:
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$$
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\begin{array} { c } { { p ^ { c } = \mathrm { A T T N } ( \{ e _ { c } \} , q ) , } } \\ { { p ^ { \mathrm { c o p y } } = p ^ { r } \otimes p ^ { c } , } } \end{array}
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$$
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where $p ^ { c }$ is a probability distribution over columns, whereas $p ^ { r }$ is a probability distribution over rows. In order to compute a matrix with the probability of copying each cell, we simply compute the outer product $p ^ { \mathrm { c o p y } } = p ^ { r } \otimes p ^ { c }$ .
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Objective: As we treat $z _ { i }$ as a latent variable, we wish to maximize the marginal probability of the sequence $y _ { i }$ over all possible values of $z _ { i }$ . Thus, our objective function is defined as:
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+
$$
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p ( y _ { i , v } | s _ { i , v } ) = p ^ { \mathrm { v o c a b } } p ( 0 | s _ { i , v } ) + p ^ { \mathrm { c o p y } } p ( 1 | s _ { i , v } ) = p ^ { \mathrm { v o c a b } } ( 1 - p ( 1 | s _ { i , v } ) ) + p ^ { \mathrm { c o p y } } p ( 1 | s _ { i , v } ) .
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$$
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The model can also be trained in a fully supervised fashion, if $z _ { i , v }$ is observed. In such cases, we simply maximize the likelihood of $p ( z _ { i , v } | s _ { i , v } )$ , based on the observations, rather than using the marginal probability over $z _ { i , v }$ .
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# 2.2 RECIPE GENERATION
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Table 3: Ingredients and recipe for Spinach and Banana Power Smoothie.
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<table><tr><td>ingredients</td><td>recipe</td></tr><tr><td>1 cup plain soy milk 3/4 cup packed fresh spinach leaves 1 large banana, sliced</td><td>Blend soy milk and spinach leaves together in a blender until smooth. Add banana and pulse until thoroughly blended.</td></tr></table>
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Next, we consider the task of recipe generation conditioning on the ingredient lists. In this task, we must generate the recipe from a list of ingredients. Table. 3 illustrates the ingredient list and recipe for Spinach and Banana Power Smoothie. We can see that the ingredients soy milk, spinach leaves, and banana occur in the recipe.
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Figure 4: Recipe pointer
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Let the ingredients of a recipe be $X ~ = ~ \{ x _ { i } \} _ { i = 1 } ^ { T }$ and each ingredient contains $L$ tokens $\begin{array} { r l } { x _ { i } } & { { } = } \end{array}$ $\{ x _ { i j } \} _ { j = 1 } ^ { L }$ . The corresponding recipe is $y = \{ y _ { v } \} _ { v = 1 } ^ { K }$ . We first use a LSTM to encode each ingredient:
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+
$$
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h _ { i , j } = \mathrm { L S T M } _ { \mathrm { E } } ( W _ { E } x _ { i j } , h _ { i , j - 1 } ) \quad \forall i .
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$$
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Then, we sum the resulting state of each ingredient to obtain the starting LSTM state of the decoder. Once again we use an attention based decoder:
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$$
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\begin{array} { c } { { s _ { v } = \mathrm { L S T M } _ { \mathrm { D } } \displaystyle ( s _ { v - 1 } , d _ { v - 1 } , W _ { \mathrm { E } } y _ { v - 1 } ) , } } \\ { { p _ { v } ^ { \mathrm { c o p y } } = \mathrm { A T T N } \displaystyle ( \{ \left\{ h _ { i , j } \right\} _ { i = 1 } ^ { T } \} _ { j = 1 } ^ { L } , s _ { v } ) , } } \\ { { d _ { v } = \displaystyle \sum _ { i j } p _ { v , i , j } h _ { i , j } , } } \\ { { p ( z _ { v } | s _ { v } ) = \mathrm { s i g m o i d } ( W [ s _ { v } , d _ { v } ] ) , } } \\ { { p _ { v } ^ { \mathrm { v o c a b } } = \mathrm { s o f t m a x } ( W [ s _ { v } , d _ { v } ] ) . } } \end{array}
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+
$$
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Similar to the previous task, the decision to copy from the ingredient list or generate a new
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word from the softmax is performed using a switch, denoted as $p ( z _ { v } | s _ { v } )$ . We can obtain a copying each of the words in the ingredients by computing in the attention mechanism. For training, we optimize the $p _ { v } ^ { \mathrm { c o p y } } =$
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$\mathsf { \bar { A } T T N } ( \{ \{ \bar { h } _ { i , j } \} _ { i = 1 } ^ { T } \} _ { j = 1 } ^ { L } , s _ { v } )$
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likelihood function employed in the previous task.
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# 2.3 COREFERENCE BASED LANGUAGE MODEL
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Finally, we build a language model that uses coreference links to point to previous words. Before generating a word, we first make the decision on whether it is an entity mention. If so, we decide which entity this mention belongs to, then we generate the word based on that entity. Denote the document as $X = \{ x _ { i } \} _ { i = 1 } ^ { L }$ , and the entities are $E = \{ e _ { i } \} _ { i = 1 } ^ { N }$ , each entity has $M _ { i }$ mentions, $e _ { i } =$ $\{ m _ { i j } \} _ { j = 1 } ^ { M _ { i } }$ , such that $\{ x _ { m _ { i j } } \} _ { j = 1 } ^ { M _ { i } }$ refer to the same entity. We use a LSTM to model the document, the hidden state of each token is $h _ { i } = \mathrm { L S T M } ( W _ { E } x _ { i } , h _ { i - 1 } )$ . We use a set $h ^ { e } = \{ h _ { 0 } ^ { e } , h _ { 1 } ^ { e } , . . . , h _ { M } ^ { e } \}$ to keep track of the entity states, where $h _ { j } ^ { e }$ is the state of entity $j$ .
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um and $[ \mathrm { I l } _ { 1 }$ think that is whats - Go ahead [Linda]2. Well and thanks goes to $[ \mathrm { y o u l } _ { 1 }$ and to [the media]3 to help $[ \mathrm { u s } ] _ { 4 } . . . \mathrm { S o } [ \mathrm { o u r } ] _ { 4 }$ hat is off to all of [you]5...
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Figure 5: Coreference based language model, example taken from Wiseman et al. (2016).
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Word generation: At each time step before generating the next word, we predict whether the word is an entity mention:
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$$
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\begin{array} { r l r } { { p ^ { \mathrm { c o r e f } } ( v _ { i } | h _ { i - 1 } , h ^ { e } ) = \mathrm { A T T N } ( h ^ { e } , h _ { i - 1 } ) , } } \\ & { } & { d _ { i } = \sum _ { v _ { i } } p ( v _ { i } ) h _ { v _ { i } } ^ { e } } \\ & { } & { p ( z _ { i } | h _ { i - 1 } ) = \mathrm { s i g m o i d } ( W [ d _ { i } , h _ { i - 1 } ] ) , } \end{array}
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$$
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where $z _ { i }$ denotes whether the next word is an entity and if yes $v _ { i }$ denotes which entity the next word corefers to. If the next word is an entity mention, then $p ( x _ { i } | v _ { i } , h _ { i - 1 } , h ^ { e } ) =$ softmax $( W _ { 1 } \operatorname { t a n h } ( W _ { 2 } [ h _ { v _ { i } } ^ { e } , h _ { i - 1 } ] ) )$ else $p ( x _ { i } | h _ { i - 1 } ) = \mathrm { s o f t m a x } ( W _ { 1 } h _ { i - 1 } )$ ,
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$$
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p ( x _ { i } | x _ { < i } ) = \left\{ \begin{array} { l l } { p ( x _ { i } | h _ { i - 1 } ) p ( z _ { i } | h _ { i - 1 } , h ^ { e } ) } & { \quad \mathrm { i f } \quad z _ { i } = 0 . } \\ { p ( x _ { i } | v _ { i } , h _ { i - 1 } , h ^ { e } ) p ^ { \mathrm { c o r e f } } ( v _ { i } | h _ { i - 1 } , h ^ { e } ) p ( z _ { i } | h _ { i - 1 } , h ^ { e } ) } & { \quad \mathrm { i f } \quad z _ { i } = 1 . } \end{array} \right.
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$$
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Entity state update: We update the entity state $h ^ { e }$ at each time step. In the beginning, $h ^ { e } = \{ h _ { 0 } ^ { e } \}$ , $h _ { 0 } ^ { e }$ denotes the state of an virtual empty entity and is a learnable variable. If $z _ { i } = 1$ and $v _ { i } = 0$ , then it indicates the next word is a new entity mention, then in the next step, we append $h _ { i }$ to $h ^ { e }$ , i.e., $h ^ { e } = \{ h ^ { e } , h _ { i } \}$ , if $e _ { i } > 0$ , then we update the corresponding entity state with the new hidden state, $h ^ { e } [ v _ { i } ] = h _ { i }$ . Another way to update the entity state is to use one LSTM to encode the mention states and get the new entity state. Here we use the latest entity mention state as the new entity state for simplicity. The detailed update process is shown in Figure 5.
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# 3 EXPERIMENTS
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# 4 DATA SETS AND PREPROCESSING
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Dialogue: We use the DSTC2 data set. We only extracted the dialogue transcript from data set. There are about 3,200 dialogues in total. Since this is a small data set, we use 5-fold cross validation and report the average result over the 5 partitions. There may be multiple tokens in each table cell, for example in Table.2, the name, address, post code and phone number have multiple tokens, we replace them with one special token. For the name, address, post code and phone number of the $j$ -th row, we replace the tokens in each cell with NAME $j$ , ADDR $j$ , POSTCODE $j$ , PHONE $j$ . If a table cell is empty, we replace it with an empty token EMPTY. We do a string match in the transcript and replace the corresponding tokens in transcripts from the table with the special tokens.
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Each dialogue on average has 8 turns (16 sentences). We use a vocabulary size of 900, including about 400 table tokens and 500 words.
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Recipes: We crawl all recipes from www.allrecipes.com. There are about 31, 000 recipes in total, and every recipe has a ingredient list and a corresponding recipe. We exclude the recipes that have less than 10 tokens or more than 500 tokens, those recipes take about $0 . 1 \%$ of all data set. On average each recipe has 118 tokens and 9 ingredients. We random shuffle the whole data set and take $80 \%$ as training and $10 \%$ for validation and test. We use a vocabulary size of 10,000 in the model.
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Coref LM: We use the Xinhua News data set from Gigaword Fifth Edition and sample 100,000 documents from it that has length in range from 100 to 500. Each document has on average 234 tokens, so there are 23 million tokens in total. We use a tool to annotate all the entity mentions and use the annotation in the training. We take $80 \%$ as training and $10 \%$ as validation and test respectively. We ignore the entities that have only one mention and for the mentions that have multiple tokens, we take the token that is most frequent in the all the mentions for this entity. After the preprocessing, tokens that are entity mentions take about $10 \%$ of all tokens. We use a vocabulary size of 50,000 in the model.
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# 4.1 MODEL TRAINING AND EVALUATION
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We train all models with simple stochastic gradient descent with clipping. We use a one-layer LSTM for all RNN components. Hyper-parameters are selected using grid search based on the validation set. We use dropout after the input embedding and LSTM output. The learning rate is selected from [0.1, 0.2, 0.5, 1], maximum gradient norm is selected from [1, 2, 5, 10] and drop ratio is selected from [0.2, 0.3, 0.5]. The batch size and LSTM dimension size is slightly different for different tasks so as to make the model fit into memory. The number of epochs to train are different for each task and we drop the learning rate after reaching a given number of epochs. We report the per-word perplexity for all tasks, specifically, we report the perplexity of all words, words that can be generated from reference and non-reference words. For recipe generation, we also generate the recipe using beam size of 10 and evaluate the generated recipe with BLEU.
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<table><tr><td>model</td><td>all</td><td>table</td><td>table oov</td><td>word</td></tr><tr><td>seq2seq</td><td>1.35±0.01</td><td>4.98±0.38</td><td>1.99E7±7.75E6</td><td>1.23±0.01</td></tr><tr><td>table attn</td><td>1.37±0.01</td><td>5.09±0.64</td><td>7.91E7±1.39E8</td><td>1.24±0.01</td></tr><tr><td>table pointer</td><td>1.33±0.01</td><td>3.99±0.36</td><td>1360 ± 2600</td><td>1.23±0.01</td></tr><tr><td>table latent</td><td>1.36±0.01</td><td>4.99±0.20</td><td>3.78E7±6.08E7</td><td>1.24±0.01</td></tr><tr><td colspan="5">+ sentence attn</td></tr><tr><td>seq2seq</td><td>1.28±0.01</td><td>3.31±0.21</td><td>2.83E9±4.69E9</td><td>1.19±0.01</td></tr><tr><td>table attn</td><td>1.28±0.01</td><td>3.17±0.21</td><td>1.67E7±9.5E6</td><td>1.20±0.01</td></tr><tr><td>table pointer</td><td>1.27±0.01</td><td>2.99±0.19</td><td>82.86±110</td><td>1.20±0.01</td></tr><tr><td>table latent</td><td>1.28±0.01</td><td>3.26±0.25</td><td>1.27E7±1.41E7</td><td>1.20±0.01</td></tr></table>
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Table 4: Dialogue perplexity results. (All means all tokens, table means tokens from table, table oov denotes table tokens that does not appear in the training set, word means non-table tokens). sentence attn denotes we use attention mechanism over tokens from past turn. Table pointer and table latent differs in that table pointer, we provide supervised signal on when to generate a table token, while in table latent it is a latent decision.
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<table><tr><td rowspan="3">model</td><td colspan="4">val</td><td colspan="4">test</td></tr><tr><td colspan="3">ppl</td><td rowspan="2">BLEU</td><td colspan="3">ppl</td><td rowspan="2">BLEU</td></tr><tr><td>all</td><td>ing</td><td>word</td><td>all</td><td>ing</td><td>word</td></tr><tr><td>seq2seq</td><td>5.60</td><td>11.26</td><td>5.00</td><td>14.07</td><td>5.52</td><td>11.26</td><td>4.91</td><td>14.39</td></tr><tr><td>attn</td><td>5.25</td><td>6.86</td><td>5.03</td><td>14.84</td><td>5.19</td><td>6.92</td><td>4.95</td><td>15.15</td></tr><tr><td>pointer</td><td>5.15</td><td>5.86</td><td>5.04</td><td>15.06</td><td>5.11</td><td>6.04</td><td>4.98</td><td>15.29</td></tr><tr><td>latent</td><td>5.02</td><td>5.10</td><td>5.01</td><td>14.87</td><td>4.97</td><td>5.19</td><td>4.94</td><td>15.41</td></tr></table>
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Table 5: Recipe result, evaluated in perplexity and BLEU score. ing denotes tokens from recipe that appear in ingredients.
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<table><tr><td rowspan="2">model</td><td colspan="3">val</td><td colspan="3">test</td></tr><tr><td>all</td><td>entity</td><td>word</td><td>all</td><td>entity</td><td>word</td></tr><tr><td>lm</td><td>33.08</td><td>44.52</td><td>32.04</td><td>33.08</td><td>43.86</td><td>32.10</td></tr><tr><td>pointer</td><td>32.57</td><td>32.07</td><td>32.62</td><td>32.62</td><td>32.07</td><td>32.69</td></tr><tr><td>pointer +init</td><td>30.43</td><td>28.56</td><td>30.63</td><td>30.42</td><td>28.56</td><td>30.66</td></tr></table>
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Table 6: Coreference based LM. pointer $^ +$ init means we initialize the model with the LM weights.
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# 4.2 RESULTS AND ANALYSIS
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The results for dialogue, recipe generation and coref language model are shown in Table 4, 5 and 6 respectively. We can see from Table 4 that models that condition on table performs better in predicting table tokens in general. Table pointer has the lowest perplexity for token in the table. Since the table token appears rarely in the dialogue, the overall perplexity does not differ much and the non-table tokens perplexity are similar. With attention mechanism over the table, the perplexity of table token improves over basic seq2seq model, but not as good as directly pointing to cells in the table. As expected, using sentence attention improves significantly over models without sentence attention. Surprisingly, table latent performs much worse than table pointer. We also measure the perplexity of table tokens that appear only in test set. For models other than table pointer, because the tokens never appear in training set, the perplexity is quite high, while table pointer can predict these tokens much more accurately. The recipe results in Table 5 in general follows that findings from the dialogue. But the latent model performs better than pointer model since that tokens in ingredients that match with recipe does not necessarily come from the ingredients. Imposing a supervised signal will give wrong information to the model and hence make the result worse. Hence with latent decision, the model learns to when to copy and when to generate it from the vocabulary. The coref LM results are shown in Table 6. We find that coref based LM performs much better on the entities perplexities, but however is a little bit worse than for non-entity words. We found it is an optimization problem and perhaps the model is stuck in local optimum. So we initialize the pointer model with the weights learned from LM, the pointer model performs better than LM both for entity perplexity and non-entity words perplexity.
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# 5 RELATED WORK
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Recently, there has been great progresses in modeling languages based on neural network, including language modeling (Mikolov et al., 2010; Jozefowicz et al., 2016), machine translation (Sutskever et al., 2014; Bahdanau et al., 2014), question answering (Hermann et al., 2015) etc. Based on the success of seq2seq models, neural networks are applied in modeling chit-chat dialogue (Li et al., 2016; Vinyals & Le, 2015; Sordoni et al., 2015; Serban et al., 2016; Shang et al., 2015) and task oriented dialogue (Wen et al., 2015; Bordes & Weston, 2016; Williams & Zweig, 2016; Wen et al., 2016). Most of the chit-chat neural dialogue models are simply applying the seq2seq models. For the task oriented dialogues, most of them embed the seq2seq model in traditional dialogue systems, in which the table query part is not differentiable. while our model queries the database directly. Recipe generation was proposed in (Kiddon et al., 2016). Their model extents previous work on attention models (Allamanis et al., 2016) to checklists, whereas our work models explicit references to those checklists. Context dependent language models (Mikolov et al., 2010; Ji et al., 2015; Wang & Cho, 2015) are proposed to capture long term dependency of text. There are also lots of works on coreference resolution (Haghighi & Klein, 2010; Wiseman et al., 2016). We are the first to combine coreference with language modeling, to the best of our knowledge. Much effort has been invested in embedding a copying mechanism for neural models (Gulc¸ehre et al. ¨ , 2016; Gu et al., 2016; Ling et al., 2016). In general, a gating mechanism is employed to combine the softmax over observed words and a pointer network (Vinyals et al., 2015). These gates can be trained either by marginalizing over both outcomes, or using heuristics (e.g. copy low frequency words). Our models are similar to models proposed in (Ahn et al., 2016; Merity et al., 2016), where the generation of each word can be conditioned on a particular entry in knowledge lists and previous words. In our work, we describe a model with broader applications, allowing us to condition, on databases, lists and dynamic lists.
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# 6 CONCLUSION
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We introduce reference-aware language models which explicitly model the decision of from where to generate the token at each step. Our model can also learns the decision by treating it as a latent variable. We demonstrate on three tasks, table based dialogue modeling, recipe generation and coref based LM, that our model performs better than attention based model, which does not incorporate this decision explicitly. There are several directions to explore further based on our framework. The current evaluation method is based on perplexity and BLEU. In task oriented dialogues, we can also try human evaluation to see if the model can reply users’ query accurately. It is also interesting to use reinforcement learning to learn the actions in each step.
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# REFERENCES
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Miltiadis Allamanis, Hao Peng, and Charles A. Sutton. A convolutional attention network for extreme summarization of source code. CoRR, abs/1602.03001, 2016. URL http://arxiv. org/abs/1602.03001.
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Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. CoRR, abs/1409.0473, 2014. URL http://arxiv.org/ abs/1409.0473.
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Antoine Bordes and Jason Weston. Learning end-to-end goal-oriented dialog. arXiv preprint arXiv:1605.07683, 2016.
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C¸ aglar Gulc¸ehre, Sungjin Ahn, Ramesh Nallapati, Bowen Zhou, and Yoshua Bengio. Pointing ¨ the unknown words. CoRR, abs/1603.08148, 2016. URL http://arxiv.org/abs/1603. 08148.
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# A VISUALIZATION
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M: Hello , welcome to the Cambridge restaurant system ? You can ask for restaurants by area , price range or food type . How may I help you ?
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U: and expensive restaurant that serves tuscan food
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M: Sorry there is no tuscan restaurant in the expensive price range
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U: french food
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M: cote is a great restaurant serving french food and it is in the expensive price range
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U: may i have the phone number
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M: The phone number of cote is 01223 311053 .
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U: thank you good bye
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M: Hello , welcome to the Cambridge restaurant system ? You can ask for restaurants by area , price range or food type . How may I help you ?
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U: cheap restaurant
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M: What kind of food would you like ?
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U: in the west part of town
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M: la margherita is a nice restaurant in the west of town in the cheap price range
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U: address
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M: Sure , la margherita is on 15 Magdalene Street City Centre
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U: thank you
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M: la margherita is a nice restaurant in the west of town in the cheap price range
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+
U: good bye
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+
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(a) Dialogue script
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+
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| 307 |
+
<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICE RANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>charlie chan</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>east</td><td rowspan=1 colspan=1>Regent Street City Cen-tre</td><td rowspan=1 colspan=1>C.B 2, 1 D.B</td><td rowspan=1 colspan=1>01223 361763</td></tr><tr><td rowspan=1 colspan=1>chiquito restau-rant bar</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>mexican</td><td rowspan=1 colspan=1>south</td><td rowspan=1 colspan=1>2G Cambridge LeisurePark Cherry HintonRoad Cherry Hinton</td><td rowspan=1 colspan=1>C.B 1,7D.Y</td><td rowspan=1 colspan=1>01223 400170</td></tr><tr><td rowspan=1 colspan=1>city stop</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>food</td><td rowspan=1 colspan=1>north</td><td rowspan=1 colspan=1>Cambridge City Foot-ball Club Milton RoadChesterton</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 363270</td></tr><tr><td rowspan=1 colspan=1>clowns cafe</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>italian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>C.B 1,1 L.N</td><td rowspan=1 colspan=1>01223 355711</td></tr><tr><td rowspan=1 colspan=1>cocum</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1>71 CastleStreet CityCentre</td><td rowspan=1 colspan=1>C.B 3,0 A.H</td><td rowspan=1 colspan=1>01223 366668</td></tr><tr><td rowspan=1 colspan=1>cote</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>french</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>Bridge Street City Cen-tre</td><td rowspan=1 colspan=1>C.B 2, 1U.F</td><td rowspan=1 colspan=1>01223 311053</td></tr><tr><td rowspan=1 colspan=1> curry garden</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>106 Regent Street CityCentre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 302330</td></tr><tr><td rowspan=1 colspan=1>curry king</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>5Jordans Yard BridgeStreet City Centre</td><td rowspan=1 colspan=1>C.B 1,2 B.D</td><td rowspan=1 colspan=1>01223 324351</td></tr><tr><td rowspan=1 colspan=1>curry prince</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>east</td><td rowspan=1 colspan=1>451 Newmarket RoadFen Ditton</td><td rowspan=1 colspan=1>C.B 5, 8 J.J</td><td rowspan=1 colspan=1>01223 566388</td></tr></table>
|
| 308 |
+
|
| 309 |
+
(b) Attention heat map: cote is a great restaurant serving french food and it is in the expensive price range.
|
| 310 |
+
Table 7: Dialogue visualization 1
|
| 311 |
+
|
| 312 |
+
<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICERANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>charlie chan</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>east</td><td rowspan=1 colspan=1>Regent Street City Cen-tre</td><td rowspan=1 colspan=1>C.B 2,1 D.B</td><td rowspan=1 colspan=1>01223 361763</td></tr><tr><td rowspan=1 colspan=1>chiquito restau-rant bar</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>mexican</td><td rowspan=1 colspan=1>south</td><td rowspan=1 colspan=1>2G Cambridge LeisurePark Cherry HintonRoad CherryHinton</td><td rowspan=1 colspan=1>C.B 1,7D.Y</td><td rowspan=1 colspan=1>01223 400170</td></tr><tr><td rowspan=1 colspan=1>city stop</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>food</td><td rowspan=1 colspan=1>north</td><td rowspan=1 colspan=1>Cambridge City Foot-ball Club Milton RoadChesterton</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 363270</td></tr><tr><td rowspan=1 colspan=1>clowns cafe</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>italian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>C.B1, 1 L.N</td><td rowspan=1 colspan=1>01223 355711</td></tr><tr><td rowspan=1 colspan=1>cocum</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1>71 CastleStreet CityCentre</td><td rowspan=1 colspan=1>C.B 3,0 A.H</td><td rowspan=1 colspan=1>01223 366668</td></tr><tr><td rowspan=1 colspan=1>cote</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>french</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>Bridge Street City Cen-tre</td><td rowspan=1 colspan=1>C.B 2,1 U.F</td><td rowspan=1 colspan=1>01223 311053</td></tr><tr><td rowspan=1 colspan=1>curry garden</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>106 Regent Street CityCentre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223302330</td></tr><tr><td rowspan=1 colspan=1> curry king</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>5Jordans Yard BridgeStreet City Centre</td><td rowspan=1 colspan=1>C.B1,2 B.D</td><td rowspan=1 colspan=1>01223 324351</td></tr><tr><td rowspan=1 colspan=1>curry prince</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>east</td><td rowspan=1 colspan=1>451 Newmarket RoadFen Ditton</td><td rowspan=1 colspan=1>C.B 5, 8 J.J</td><td rowspan=1 colspan=1>01223 566388</td></tr></table>
|
| 313 |
+
|
| 314 |
+
(c) Attention heap map: The phone number of cote is 01223 311053 .
|
| 315 |
+
|
| 316 |
+
(a) Dialogue script
|
| 317 |
+
|
| 318 |
+
<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICE RANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>india house</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1>31Newnham RoadNewnham</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 461661</td></tr><tr><td rowspan=1 colspan=1>j restaurant</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>oriental</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>86Regent Street CityCentre</td><td rowspan=1 colspan=1>C.B 2,1 D.P</td><td rowspan=1 colspan=1>01223 307581</td></tr><tr><td rowspan=1 colspan=1>jinlingnoodlebar</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>11 Peas Hill City Cen-tre</td><td rowspan=1 colspan=1>C.B 2, 3 P.P</td><td rowspan=1 colspan=1>01223 566188</td></tr><tr><td rowspan=1 colspan=1>kohinoor</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>74 Mill Road City Cen-tre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 323639</td></tr><tr><td rowspan=1 colspan=1>kymmoy</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>oriental</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1> 52 Mill Road City Cen-tre</td><td rowspan=1 colspan=1>C.B 1,2 A.S</td><td rowspan=1 colspan=1>01223 311911</td></tr><tr><td rowspan=1 colspan=1>la margherita</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>italian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1>15MagdaleneStreetCity Centre</td><td rowspan=1 colspan=1>C.B 3,0 A.F</td><td rowspan=1 colspan=1>01223 315232</td></tr><tr><td rowspan=1 colspan=1>la mimosa</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>mediterranean</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>ThompsonsLane FenDitton</td><td rowspan=1 colspan=1>C.B 5,8 A.Q</td><td rowspan=1 colspan=1>01223 362525</td></tr><tr><td rowspan=1 colspan=1>la raza</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>spanish</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>4-6Rose Crescent</td><td rowspan=1 colspan=1>C.B 2, 3L.L</td><td rowspan=1 colspan=1>01223 464550</td></tr><tr><td rowspan=1 colspan=1>la tasca</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>spanish</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>14 -16 Bridge Street</td><td rowspan=1 colspan=1>C.B 2,1U.F</td><td rowspan=1 colspan=1>01223464630</td></tr><tr><td rowspan=1 colspan=1>lan hong house</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>12 Norfolk Street CityCentre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 350420</td></tr></table>
|
| 319 |
+
|
| 320 |
+
(b) Attention heat map: la margherita is a nice restaurant in the west of town in the cheap price range
|
| 321 |
+
Table 8: Dialogue visualization 2
|
| 322 |
+
|
| 323 |
+
<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICE RANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>india house</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1>311Newnham RoadNewnham</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 461661</td></tr><tr><td rowspan=1 colspan=1>jrestaurant</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>oriental</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>86RegentStreet CityCentre</td><td rowspan=1 colspan=1>C.B 2, 1 D.P</td><td rowspan=1 colspan=1>01223 307581</td></tr><tr><td rowspan=1 colspan=1> jinlingnoodlebar</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>11 Peas Hill City Cen-tre</td><td rowspan=1 colspan=1>C.B 2, 3 P.P</td><td rowspan=1 colspan=1>01223 566188</td></tr><tr><td rowspan=1 colspan=1>kohinoor</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>74 Mill Road City Cen-tre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 323639</td></tr><tr><td rowspan=1 colspan=1>kymmoy</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>oriental</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1> 52 Mill Road City Cen-tre</td><td rowspan=1 colspan=1>C.B 1,2 A.S</td><td rowspan=1 colspan=1>01223 311911</td></tr><tr><td rowspan=1 colspan=1>la margherita</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>italian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1> 15 MagdaleneStreetCity Centre</td><td rowspan=1 colspan=1>C.B 3,0 A.F</td><td rowspan=1 colspan=1>01223 315232</td></tr><tr><td rowspan=1 colspan=1>la mimosa</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>mediterranean</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>ThompsonsLane FenDitton</td><td rowspan=1 colspan=1>C.B 5, 8 A.Q</td><td rowspan=1 colspan=1>01223 362525</td></tr><tr><td rowspan=1 colspan=1>la raza</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>spanish</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>4 -6 Rose Crescent</td><td rowspan=1 colspan=1>C.B 2, 3 L.L</td><td rowspan=1 colspan=1>01223 464550</td></tr><tr><td rowspan=1 colspan=1>la tasca</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>spanish</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>14-16 Bridge Street</td><td rowspan=1 colspan=1>C.B 2, 1 U.F</td><td rowspan=1 colspan=1>01223 464630</td></tr><tr><td rowspan=1 colspan=1>lan hong house</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>12 Norfolk Street CityCentre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 350420</td></tr></table>
|
| 324 |
+
|
| 325 |
+
(c) Attention heap map: Sure , la margherita is on 15 Magdalene Street City Centre.
|
| 326 |
+
|
| 327 |
+
|
| 328 |
+
Figure 6: Recipe heat map example 1. The ingredient tokens appear on the left while the recipe tokens appear on the top. The first row is the $p \big ( \bar { z } _ { v } | s _ { v } \big )$ .
|
| 329 |
+
|
| 330 |
+
|
| 331 |
+
Figure 7: Recipe heat map example 2.
|
parse/train/ByG8A7cee/ByG8A7cee_content_list.json
ADDED
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "REFERENCE-AWARE LANGUAGE MODELS ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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|
| 12 |
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"page_idx": 0
|
| 13 |
+
},
|
| 14 |
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{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Zichao $\\mathbf { Y a n g ^ { 1 * } }$ ∗, Phil Blunsom2,3, Chris Dyer1,2, and Wang Ling2 1Carnegie Mellon University, 2DeepMind, and 3University of Oxford zichaoy@cs.cmu.edu, {pblunsom,cdyer,lingwang}@google.com ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
184,
|
| 19 |
+
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|
| 20 |
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730,
|
| 21 |
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|
| 22 |
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],
|
| 23 |
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"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "ABSTRACT ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
454,
|
| 31 |
+
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|
| 32 |
+
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|
| 33 |
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|
| 34 |
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|
| 35 |
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"page_idx": 0
|
| 36 |
+
},
|
| 37 |
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{
|
| 38 |
+
"type": "text",
|
| 39 |
+
"text": "We propose a general class of language models that treat reference as an explicit stochastic latent variable. This architecture allows models to create mentions of entities and their attributes by accessing external databases (required by, e.g., dialogue generation and recipe generation) and internal state (required by, e.g. language models which are aware of coreference). This facilitates the incorporation of information that can be accessed in predictable locations in databases or discourse context, even when the targets of the reference may be rare words. Experiments on three tasks show our model variants outperform models based on deterministic attention. ",
|
| 40 |
+
"bbox": [
|
| 41 |
+
233,
|
| 42 |
+
256,
|
| 43 |
+
764,
|
| 44 |
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|
| 45 |
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],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 INTRODUCTION ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
176,
|
| 54 |
+
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|
| 55 |
+
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|
| 56 |
+
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|
| 57 |
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],
|
| 58 |
+
"page_idx": 0
|
| 59 |
+
},
|
| 60 |
+
{
|
| 61 |
+
"type": "text",
|
| 62 |
+
"text": "Referring expressions (REs) in natural language are noun phrases (proper nouns, common nouns, and pronouns) that identify objects, entities, and events in an environment. REs occur frequently and they play a key role in communicating information efficiently. While REs are common, previous works neglect to model REs explicitly, either treating REs as ordinary words in the model or replacing them with special tokens. Here we propose a language modeling framework that explicitly incorporates reference decisions. ",
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
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|
| 66 |
+
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|
| 67 |
+
523
|
| 68 |
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],
|
| 69 |
+
"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "In Figure 1 we list examples of REs in the context of the three tasks that we consider in this work. Firstly, reference to a database is crucial in many applications. One example is in task oriented dialogue where access to a database is necessary to answer a user’s query (Young et al., 2013; Li et al., 2016; Vinyals & Le, 2015; Wen et al., 2015; Sordoni et al., 2015; Serban et al., 2016; Bordes & Weston, 2016; Williams & Zweig, 2016; Shang et al., 2015; Wen et al., 2016). Here we consider the domain of restaurant recommendation where a system refers to restaurants (name) and their attributes (address, phone number etc) in its responses. When the system says “the nirala is a nice restaurant”, it refers to the restaurant name the nirala from the database. Secondly, many models need to refer to a list of items (Kiddon et al., 2016; Wen et al., 2015). In the task of recipe generation from a list of ingredients (Kiddon et al., 2016), the generation of the recipe will frequently reference these items. As shown in Figure 1, in the recipe “Blend soy milk and . . . ”, soy milk refers to the ingredient summaries. Finally, we address references within a document (Mikolov et al., 2010; Ji et al., 2015; Wang & Cho, 2015), as the generation of words will ofter refer to previously generated words. For instance the same entity will often be referred to throughout a document. In Figure 1, the entity you refers to I in a previous utterance. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
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|
| 76 |
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|
| 77 |
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|
| 78 |
+
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|
| 79 |
+
],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "In this work we develop a language model that has a specific module for generating REs. A series of latent decisions (should I generate a RE? If yes, which entity in the context should I refer to? How should the RE be rendered?) augment a traditional recurrent neural network language model and the two components are combined as a mixture model. Selecting an entity in context is similar to familiar models of attention (Bahdanau et al., 2014), but rather than being a deterministic function that reweights representations of elements in the context, it is treated as a distribution over contextual elements which are stochastically selected and then copied or, if the task warrants it, transformed (e.g., a pronoun rather than a proper name is produced as output). Two variants are possible for updating the RNN state: one that only looks at the generated output form; and a second that looks at values of the latent variables. The former admits trivial unsupervised learning, latent decisions are conditionally independent of each other given observed context, whereas the latter enables more expressive models that can extract information from the entity that is being referred to. In each of the three tasks, we demonstrate our reference aware model’s efficacy in evaluations against models that do not explicitly include a reference operation. ",
|
| 85 |
+
"bbox": [
|
| 86 |
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|
| 87 |
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| 88 |
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| 89 |
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|
| 90 |
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],
|
| 91 |
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"page_idx": 0
|
| 92 |
+
},
|
| 93 |
+
{
|
| 94 |
+
"type": "image",
|
| 95 |
+
"img_path": "images/b3c493786dc3a9b650fc7c012d8f1e8c16fa322597d6a17781862a6e4457f828.jpg",
|
| 96 |
+
"image_caption": [
|
| 97 |
+
"Figure 1: Reference-aware language models. "
|
| 98 |
+
],
|
| 99 |
+
"image_footnote": [],
|
| 100 |
+
"bbox": [
|
| 101 |
+
281,
|
| 102 |
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|
| 103 |
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|
| 104 |
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|
| 105 |
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],
|
| 106 |
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"page_idx": 1
|
| 107 |
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},
|
| 108 |
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{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "",
|
| 111 |
+
"bbox": [
|
| 112 |
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174,
|
| 113 |
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|
| 114 |
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|
| 115 |
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|
| 116 |
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],
|
| 117 |
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"page_idx": 1
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "Our contributions are as follows: ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
174,
|
| 124 |
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|
| 125 |
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390,
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| 126 |
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| 127 |
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],
|
| 128 |
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"page_idx": 1
|
| 129 |
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},
|
| 130 |
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{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "• We propose a general framework to model reference in language and instantiate it in the context of dialogue modeling, recipe generation and coreference based language models. • We build three data sets to test our models. There lack existing data sets that satisfy our need, so we build these data sets ourselves. These data sets are either built on top existing data set (we constructed the table for DSTC2 data set for dialogue evaluation), crawled from websites (we crawled all recipes in www.allrecipes.com) or annotated with NLP tools (we annotate the coreference with Gigaword corpus for our evaluation). • We perform comprehensive evaluation of our models on the three data sets and verify our models perform better than strong baselines. ",
|
| 133 |
+
"bbox": [
|
| 134 |
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| 135 |
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| 136 |
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| 137 |
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| 138 |
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],
|
| 139 |
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"page_idx": 1
|
| 140 |
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},
|
| 141 |
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{
|
| 142 |
+
"type": "text",
|
| 143 |
+
"text": "2 REFERENCE-AWARE LANGUAGE MODELS ",
|
| 144 |
+
"text_level": 1,
|
| 145 |
+
"bbox": [
|
| 146 |
+
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|
| 147 |
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|
| 148 |
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|
| 149 |
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549
|
| 150 |
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],
|
| 151 |
+
"page_idx": 1
|
| 152 |
+
},
|
| 153 |
+
{
|
| 154 |
+
"type": "text",
|
| 155 |
+
"text": "Here we propose a general framework for reference-aware language models. ",
|
| 156 |
+
"bbox": [
|
| 157 |
+
176,
|
| 158 |
+
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|
| 159 |
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|
| 160 |
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578
|
| 161 |
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],
|
| 162 |
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"page_idx": 1
|
| 163 |
+
},
|
| 164 |
+
{
|
| 165 |
+
"type": "text",
|
| 166 |
+
"text": "We denote each document as a series of tokens $x _ { 1 } , \\ldots , x _ { L }$ , where $L$ is the number of tokens in the document. Our goal is to maximize the probabilities $p ( x _ { i } \\mid c _ { i } )$ , for each word in the document based on its previous context $c _ { i } = x _ { 1 } , \\ldots , x _ { i - 1 }$ . In contrast to traditional neural language models, we introduce a variable at each position $z _ { i }$ , which controls the decision on which source $x _ { i }$ is generated from. The token conditional probably is then obtained by: ",
|
| 167 |
+
"bbox": [
|
| 168 |
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|
| 169 |
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| 170 |
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| 171 |
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| 172 |
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],
|
| 173 |
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"page_idx": 1
|
| 174 |
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},
|
| 175 |
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{
|
| 176 |
+
"type": "equation",
|
| 177 |
+
"img_path": "images/a408daa414778b14a0cab2c0d09ff2fd66785f031873300b5337363e5f169b6a.jpg",
|
| 178 |
+
"text": "$$\np ( x _ { i } \\mid c _ { i } ) = p ( x _ { i } \\mid z _ { i } , c _ { i } ) p ( z _ { i } \\mid c _ { i } ) .\n$$",
|
| 179 |
+
"text_format": "latex",
|
| 180 |
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"bbox": [
|
| 181 |
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380,
|
| 182 |
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| 183 |
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|
| 184 |
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| 185 |
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],
|
| 186 |
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"page_idx": 1
|
| 187 |
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},
|
| 188 |
+
{
|
| 189 |
+
"type": "text",
|
| 190 |
+
"text": "In dialogue modeling and recipe generation, $z _ { i }$ will simply taken on values in $\\{ 0 , 1 \\}$ . Where $z _ { i } = 1$ denotes that $x _ { i }$ is generated as a reference, either to a database entry or an item in a list. However, $z _ { i }$ can also be defined as a distribution over previous entities, allowing the model to predict $x _ { i }$ conditioned on its a previous mention word. This will be the focus of the coreference language model. When $z _ { i }$ is not observed (which it generally will not be), we will train our model to maximize the marginal probability in Eq. 1 directly. ",
|
| 191 |
+
"bbox": [
|
| 192 |
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| 193 |
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| 194 |
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| 196 |
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| 197 |
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"page_idx": 1
|
| 198 |
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},
|
| 199 |
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{
|
| 200 |
+
"type": "text",
|
| 201 |
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"text": "2.1 DIALOGUE MODEL WITH DATABASE SUPPORT ",
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"text": "We first apply our model on task-oriented dialogue systems in the domain of restaurant recommendations, and work on the data set from the second Dialogue State Tracking Challenge (DSTC2) (Henderson et al., 2014). Table. 1 is one example dialogue from this dataset. ",
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"text": "We can observe from this example, users get recommendations of restaurants based on queries that specify the area, price and food type of the restaurant. We can support the system’s decisions by incorporating a mechanism that allows the model to query the database allowing the model to find restaurants that satisfy the users queries. Thus, we crawled TripAdvisor for restaurants in the ",
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"text": "M: Hello , welcome to the Cambridge restaurant system? You can ask for restaurants by area, price range or food type . How may I help you? U: i want a moderately priced restaurant in the north town M: the nirala is a nice restaurant in the north of town in the moderate price range U: what is the address M: Sure , the nirala is on 7 Milton Road Chesterton ",
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"img_path": "images/801ffef07eb629bad790ce259b827d304563c3a93b5991dd3a911571718eca3e.jpg",
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"table_caption": [
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"Table 1: Example dialogue, M stands for Machine and U stands for User ",
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"Table 2: Fragment of database for dialogue system. "
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"table_body": "<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICE RANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>ali baba</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>lebanese</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>59 Hills Road City Cen-tre</td><td rowspan=1 colspan=1>CB 2,1NT</td><td rowspan=1 colspan=1>01462 432565</td></tr><tr><td rowspan=1 colspan=1>thenirala</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>north</td><td rowspan=1 colspan=1>7 Milton Road Chester-ton</td><td rowspan=1 colspan=1>CB 4,1 UY</td><td rowspan=1 colspan=1>01223 360966</td></tr></table>",
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"text": "Cambridge area, where the dialog dataset was collected. Then, we remove restaurants that do not appear in the data set and create a database with 109 entries with restaurants and their attributes (e.g. food type). A sample of our database is shown in Table. 2. We can observe that each restaurant contains 6 attributes that are generally referred in the dialogue dataset. As such, if the user requests a restaurant that serves “indian” food, we wish to train a model that can search for entries whose “food” column contains “indian”. Now, we describe how we deploy a model that fulfills these requirements. ",
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"type": "text",
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"text": "2.1.1 DIALOGUE MODEL ",
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"image_caption": [
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"Figure 2: Hierarchical RNN Seq2Seq model "
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"text": "We build a model based on the hierarchical RNN model described in (Serban et al., 2016), as in dialogues, the generation of the response is not only dependent on the previous sentence, but on all sentences leading to the response. We assume that a dialogue is alternated between a machine and a user. An illustration of the model is shown in Figure 2. ",
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"text": "Consider a dialogue with $T$ turns, and the utterance from a user is denoted as $X = \\{ x _ { i } \\} _ { i = 1 } ^ { T }$ , where $i$ is the $i$ -th utterance, whereas the utterance from a machine is denoted as ${ \\cal Y } = \\{ y _ { i } \\} _ { i = 1 } ^ { T }$ , where $i$ is the $i$ -th utterance. We define $x _ { i } = \\{ x _ { i j } \\} _ { j = 1 } ^ { | x _ { i } | }$ , $y _ { i } = \\{ y _ { i v } \\} _ { v = 1 } ^ { | y _ { i } | }$ , where $x _ { i j }$ denotes the $j$ -th token in the $i$ -th utterance from the user, whereas $y _ { i v }$ denotes the $v$ -th token in the $i$ -th utterance from the machine. Finally, $\\left| x _ { i } \\right|$ and $| y _ { i } |$ denote the number of tokens in the user and machine utterances, respectively. The dialogue sequence starts with machine utterance $\\left\\{ y _ { 1 } , x _ { 1 } , y _ { 2 } , x _ { 2 } , \\dots , y _ { T } , x _ { T } \\right\\}$ . We would like to model the utterances from the machine ",
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"text": "$$\np ( y _ { 1 } , y _ { 2 } , \\ldots , y _ { T } | x _ { 1 } , x _ { 2 } , \\ldots , x _ { T } ) = \\prod _ { i } p ( y _ { i } | y _ { < i } , x _ { < i } ) = \\prod _ { i , v } p ( y _ { i , v } | y _ { i , < v } , y _ { < i } , x _ { < i } ) ,\n$$",
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"text": "where $y _ { < i }$ denotes all the utterances before $i$ and $y _ { i , < v }$ denotes the first $v - 1$ tokens in the $i$ -th utterance of the machine. A neural model is employed to predict $p ( y _ { i , v } | y _ { i , < v } , y _ { < i } , x _ { < i } )$ , which operates as follows: ",
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"text": "Sentence Encoder: We first encode previous utterances $y _ { < i }$ and $x _ { < i }$ into continuous space by generatistate mploying a LSTM encodand apply the recursion $x _ { i }$ , and sta, where he initial LSTMdenotes a word $h _ { i , 0 } ^ { x }$ $h _ { i , j } ^ { x } = \\mathrm { L S T M } _ { \\mathrm { E } } ^ { - } ( W _ { E } x _ { i , j } , h _ { i , j - 1 } ^ { x } )$ $W _ { E } x _ { i , j }$ ",
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"text": "embedding lookup for the token $x _ { i , j }$ , and $\\mathbf { L S T M _ { E } }$ denotes the LSTM transition function described in Hochreiter & Schmidhuber (1997). The representation of the user utterance is represented by the final LSTM state $h _ { i } ^ { x } \\ = \\ h _ { i , | x _ { i } | } ^ { x }$ The same process is applied to obtain the machine utterance representation $h _ { i } ^ { y } = h _ { i , | y _ { i } | } ^ { y }$ hyi,|yi| ",
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"text": "Turn Encoder: Then, combine all the representations of all the utterances with a second LSTM, which encodes the sequence $\\{ h _ { 1 } ^ { y } , h _ { 1 } ^ { x } , . . . , h _ { i } ^ { \\bar { y } } , h _ { i } ^ { x } \\}$ into a continuous vector. Once again, we start with an initial state $u _ { 0 }$ and feed each of the utterance representation to obtain the following LSTM state, until the final state is obtained. For simplicity, we shall refer to this as $u _ { i }$ , which can be seen as the hierarchical encoding of the previous $i$ utterances. ",
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"text": "Seq2Seq Decoder: As for decoding, in order to generate each utterance $y _ { i }$ , we can feed $u _ { i - 1 }$ into the decoder LSTM as the initial state $s _ { i , 0 } = u _ { i - 1 }$ and decode each token in $y _ { i }$ . Thus, we can express the decoder as: ",
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"text": "$$\n\\begin{array} { r l } & { s _ { i , v } ^ { y } = \\mathrm { L S T M } _ { \\mathrm { D } } ( W _ { E } y _ { i , v - 1 } , s _ { i , v - 1 } ) , } \\\\ & { p _ { i , v } ^ { y } = \\mathrm { s o f t m a x } ( W s _ { i , v } ^ { y } ) , } \\end{array}\n$$",
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"text": "where the desired probability $p ( y _ { i , v } | y _ { i , < v } , y _ { < i } , x _ { < i } )$ is expressed by $p _ { i , v } ^ { y }$ ",
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"text": "Attention based decoder: We can also incorporate the attention mechanism in our hierarchical model. An attention model builds a representation $d$ by averaging over a set of vectors $p$ . We define the attention function as $a = \\mathrm { A T T N } ( p , q )$ , where $a$ is a probability distribution over the set of vectors $p$ , conditioned on any input representation $q$ . A full description of this operation is described in (Bahon the current decoder state danau et al., 2014). Thus, for each generated token $s _ { i , v } ^ { y }$ , obtaining the attentions over input tokens from previous turn $y _ { i , v }$ , we compute the attentions $a _ { i , v }$ , conditioned $( i - 1 )$ . We denote the vector of all tokens in previous turn as $h _ { i - 1 } ^ { x , y } = [ \\{ h _ { i - 1 , j } ^ { x } \\} _ { j = 1 } ^ { | x _ { i - 1 } | } , \\{ h _ { i - 1 , v } ^ { y } \\} _ { v = 1 } ^ { | y _ { i - 1 } | } ]$ }|yi−1|] Let . $K = | h _ { i - 1 } ^ { x , y } |$ be the number of tokens in previous turn. Thus, we obtain the attention probabilities − over all previous tokens $a _ { i , v }$ as $\\mathrm { A T T N } ( s _ { i , v } ^ { y } , h _ { i - 1 } ^ { x , y } )$ . Then, the weighted sum is computed over these probabilities $\\begin{array} { r } { d _ { i , v } = \\sum _ { k \\in K } a _ { i , v , k } h _ { i - 1 , k } ^ { x , y } } \\end{array}$ , where $_ { a _ { i , v , k } }$ is the probability of aligning to the $k$ -th token from previous turn. The resulting vector $d _ { i , v }$ is used to obtain the probability of the following word $p _ { i , v } ^ { y }$ . Thus, we express the decoder as: ",
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"text": "$$\n\\begin{array} { r l } & { s _ { i , v } ^ { y } = \\mathrm { L S T M } _ { \\mathrm { D } } ( [ { W } _ { \\mathrm { E } } y _ { i , v - 1 } , d _ { i , v - 1 } ] , s _ { i , v - 1 } ) , } \\\\ & { a _ { i , v } = \\mathrm { A T T N } ( h _ { i - 1 } ^ { x , y } , s _ { i , v } ^ { y } ) , } \\\\ & { d _ { i , v } = \\displaystyle \\sum _ { k \\in K } a _ { i , v , k } h _ { i - 1 , k } ^ { x , y } , } \\\\ & { p _ { i , v } ^ { y } = \\mathrm { s o f t m a x } ( W [ s _ { i , v } ^ { y } , d _ { i , v } ] ) . } \\end{array}\n$$",
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"text": "2.1.2 INCORPORATING TABLE ATTENTION ",
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"img_path": "images/b0ecbcb0faa2ea176f8574992b4d9fd0e2656f8d2a9d4622b8cbcf038acd89ca.jpg",
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"image_caption": [
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"Figure 3: Table based decoder. "
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"text": "We now extend the attention model in order to allow the attention to be computed over a table, allowing the model to condition the generation on a database. ",
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"text": "We denote a table with $R$ rows and $C$ columns as $\\{ f _ { r , c } \\} , r \\in [ 1 , R ] , c \\in [ 1 , C ]$ , where $f _ { r , c }$ is the cell in row $r$ and column $c$ . The attribute of each column is denoted as $s _ { c }$ , where $c$ is the $c$ -th attribute. $f _ { r , c }$ and $s _ { c }$ are one-hot vector. ",
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"text": "Table Encoding: To encode the table, we build an attribute vector $g _ { c }$ for each column. For each cell $f _ { r , c }$ of the table, we concatenate it with the corresponding attribute $g _ { c }$ and then feed it through a one-layer MLP as follows: $g _ { c } = W _ { E } s _ { c }$ and then $e _ { r , c } = \\operatorname { t a n h } ( W [ W _ { E } f _ { r , c } , g _ { c } ] )$ . ",
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"text": "Table Attention: The diagram for table attention is shown in Figure 3a. The attention over cells in the table is conditioned on a given vector $q$ , similarly to the attention model for sequences $\\mathbf { A T T N } ( p , q )$ . However, rather than a sequence $p$ , we now operate over a table $f$ . Our attention model computes a attribute attention followed by row attention of the table. We first use the attention mechanism on the attributes to find out which attribute the user asks about. Suppose a user says cheap, then we should focus on the price attribute. After we get the attention probability $p ^ { a } = \\mathrm { A T T N } ( \\{ g _ { c } \\} , q )$ , over the attribute, we calculate the weighted representation for each row $\\begin{array} { r } { e _ { r } = \\sum _ { c } p _ { c } ^ { a } e _ { r c } } \\end{array}$ conditioned on $p ^ { a }$ . Then $e _ { r }$ has the price information of each row. We further use attention mechanism on $e _ { r }$ and get the probability $p ^ { r } \\bar { \\mathbf { \\Psi } } = \\mathrm { A T T N } ( \\{ e _ { r } \\} , q )$ over the rows. Then restaurants with cheap price will be picked. Then, using the probabilities $p ^ { r }$ , we compute the weighted average over the all rows $\\begin{array} { r } { e _ { c } = \\sum _ { r } p _ { r } ^ { r } e _ { r , c } } \\end{array}$ , which is used in the decoder. The detailed process is: ",
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"text": "$$\n\\begin{array} { r l r } { { p _ { a } = \\mathrm { A T T N } ( \\{ g _ { c } \\} , q ) , } } \\\\ & { } & { \\quad e _ { r } = \\displaystyle \\sum _ { c } p _ { c } ^ { a } e _ { r c } \\quad \\forall r , } \\\\ & { } & { \\quad p _ { r } = \\mathrm { A T T N } ( \\{ e _ { r } \\} , q ) , } \\\\ & { } & { \\quad e _ { c } = \\displaystyle \\sum _ { r } p _ { r } ^ { r } e _ { r , c } \\quad \\forall c . } \\end{array}\n$$",
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"text": "This is embedded in the decoder by replacing the conditioned state as the current decoder state $s _ { i , 0 } ^ { y }$ and then at each step, conditioning the prediction of ach step. The detailed diagram of table attention is s $y _ { i , v }$ on n in $\\{ e _ { c } \\}$ by using attention mechanismre 3a. ",
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"text": "2.1.3 INCORPORATING TABLE POINTER NETWORKS ",
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"text": "We now describe the mechanism used to refer to specific database entries during decoding. At each timestep, the model needs to decide whether to generate the next token from an entry of the database or from the word softmax. This is performed as follows. ",
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"text": "Pointer Switch: We use $z _ { i , v } \\in [ 0 , 1 ]$ to denote the decision of whether to copy one cell from the table. We compute this probability as follows: ",
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"text": "$$\np ( z _ { i , v } | s _ { i , v } ) = \\mathrm { s i g m o i d } ( W [ s _ { i , v } , d _ { i , v } ] ) .\n$$",
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"text": "Thus, if $z _ { i , v } = 1$ , the next token $y _ { i , v }$ will be generated from the database, whereas if $z _ { i , v } = 0$ , then the following token is generated from a softmax. We shall now describe how we generate tokens from the database. ",
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"text": "Table Pointer: If $z _ { i , v } = 1$ , the token is generated from the table. The detailed process of calculating the probability distribution over the table is shown in Figure 3b. This is similar to the attention mechanism, except that we perform a column attention to compute the probabilities of copying from each column after Equation. 5. More formally: ",
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"text": "$$\n\\begin{array} { c } { { p ^ { c } = \\mathrm { A T T N } ( \\{ e _ { c } \\} , q ) , } } \\\\ { { p ^ { \\mathrm { c o p y } } = p ^ { r } \\otimes p ^ { c } , } } \\end{array}\n$$",
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"text": "where $p ^ { c }$ is a probability distribution over columns, whereas $p ^ { r }$ is a probability distribution over rows. In order to compute a matrix with the probability of copying each cell, we simply compute the outer product $p ^ { \\mathrm { c o p y } } = p ^ { r } \\otimes p ^ { c }$ . ",
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"text": "Objective: As we treat $z _ { i }$ as a latent variable, we wish to maximize the marginal probability of the sequence $y _ { i }$ over all possible values of $z _ { i }$ . Thus, our objective function is defined as: ",
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"text": "$$\np ( y _ { i , v } | s _ { i , v } ) = p ^ { \\mathrm { v o c a b } } p ( 0 | s _ { i , v } ) + p ^ { \\mathrm { c o p y } } p ( 1 | s _ { i , v } ) = p ^ { \\mathrm { v o c a b } } ( 1 - p ( 1 | s _ { i , v } ) ) + p ^ { \\mathrm { c o p y } } p ( 1 | s _ { i , v } ) .\n$$",
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"text": "The model can also be trained in a fully supervised fashion, if $z _ { i , v }$ is observed. In such cases, we simply maximize the likelihood of $p ( z _ { i , v } | s _ { i , v } )$ , based on the observations, rather than using the marginal probability over $z _ { i , v }$ . ",
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"text": "2.2 RECIPE GENERATION ",
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"table_caption": [
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"Table 3: Ingredients and recipe for Spinach and Banana Power Smoothie. "
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"table_body": "<table><tr><td>ingredients</td><td>recipe</td></tr><tr><td>1 cup plain soy milk 3/4 cup packed fresh spinach leaves 1 large banana, sliced</td><td>Blend soy milk and spinach leaves together in a blender until smooth. Add banana and pulse until thoroughly blended.</td></tr></table>",
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"text": "Next, we consider the task of recipe generation conditioning on the ingredient lists. In this task, we must generate the recipe from a list of ingredients. Table. 3 illustrates the ingredient list and recipe for Spinach and Banana Power Smoothie. We can see that the ingredients soy milk, spinach leaves, and banana occur in the recipe. ",
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"Figure 4: Recipe pointer "
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"text": "Let the ingredients of a recipe be $X ~ = ~ \\{ x _ { i } \\} _ { i = 1 } ^ { T }$ and each ingredient contains $L$ tokens $\\begin{array} { r l } { x _ { i } } & { { } = } \\end{array}$ $\\{ x _ { i j } \\} _ { j = 1 } ^ { L }$ . The corresponding recipe is $y = \\{ y _ { v } \\} _ { v = 1 } ^ { K }$ . We first use a LSTM to encode each ingredient: ",
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"text": "$$\nh _ { i , j } = \\mathrm { L S T M } _ { \\mathrm { E } } ( W _ { E } x _ { i j } , h _ { i , j - 1 } ) \\quad \\forall i .\n$$",
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"text": "Then, we sum the resulting state of each ingredient to obtain the starting LSTM state of the decoder. Once again we use an attention based decoder: ",
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"text": "$$\n\\begin{array} { c } { { s _ { v } = \\mathrm { L S T M } _ { \\mathrm { D } } \\displaystyle ( s _ { v - 1 } , d _ { v - 1 } , W _ { \\mathrm { E } } y _ { v - 1 } ) , } } \\\\ { { p _ { v } ^ { \\mathrm { c o p y } } = \\mathrm { A T T N } \\displaystyle ( \\{ \\left\\{ h _ { i , j } \\right\\} _ { i = 1 } ^ { T } \\} _ { j = 1 } ^ { L } , s _ { v } ) , } } \\\\ { { d _ { v } = \\displaystyle \\sum _ { i j } p _ { v , i , j } h _ { i , j } , } } \\\\ { { p ( z _ { v } | s _ { v } ) = \\mathrm { s i g m o i d } ( W [ s _ { v } , d _ { v } ] ) , } } \\\\ { { p _ { v } ^ { \\mathrm { v o c a b } } = \\mathrm { s o f t m a x } ( W [ s _ { v } , d _ { v } ] ) . } } \\end{array}\n$$",
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"text": "Similar to the previous task, the decision to copy from the ingredient list or generate a new \nword from the softmax is performed using a switch, denoted as $p ( z _ { v } | s _ { v } )$ . We can obtain a copying each of the words in the ingredients by computing in the attention mechanism. For training, we optimize the $p _ { v } ^ { \\mathrm { c o p y } } =$ \n$\\mathsf { \\bar { A } T T N } ( \\{ \\{ \\bar { h } _ { i , j } \\} _ { i = 1 } ^ { T } \\} _ { j = 1 } ^ { L } , s _ { v } )$ \nlikelihood function employed in the previous task. ",
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"text": "2.3 COREFERENCE BASED LANGUAGE MODEL ",
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"text": "Finally, we build a language model that uses coreference links to point to previous words. Before generating a word, we first make the decision on whether it is an entity mention. If so, we decide which entity this mention belongs to, then we generate the word based on that entity. Denote the document as $X = \\{ x _ { i } \\} _ { i = 1 } ^ { L }$ , and the entities are $E = \\{ e _ { i } \\} _ { i = 1 } ^ { N }$ , each entity has $M _ { i }$ mentions, $e _ { i } =$ $\\{ m _ { i j } \\} _ { j = 1 } ^ { M _ { i } }$ , such that $\\{ x _ { m _ { i j } } \\} _ { j = 1 } ^ { M _ { i } }$ refer to the same entity. We use a LSTM to model the document, the hidden state of each token is $h _ { i } = \\mathrm { L S T M } ( W _ { E } x _ { i } , h _ { i - 1 } )$ . We use a set $h ^ { e } = \\{ h _ { 0 } ^ { e } , h _ { 1 } ^ { e } , . . . , h _ { M } ^ { e } \\}$ to keep track of the entity states, where $h _ { j } ^ { e }$ is the state of entity $j$ . ",
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"text": "um and $[ \\mathrm { I l } _ { 1 }$ think that is whats - Go ahead [Linda]2. Well and thanks goes to $[ \\mathrm { y o u l } _ { 1 }$ and to [the media]3 to help $[ \\mathrm { u s } ] _ { 4 } . . . \\mathrm { S o } [ \\mathrm { o u r } ] _ { 4 }$ hat is off to all of [you]5... ",
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"type": "image",
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"img_path": "images/787f483a8556a3d8a8f59733ad85ae9b906c646cdc53288cac6b92ec15023293.jpg",
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| 821 |
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"image_caption": [
|
| 822 |
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"Figure 5: Coreference based language model, example taken from Wiseman et al. (2016). "
|
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"text": "Word generation: At each time step before generating the next word, we predict whether the word is an entity mention: ",
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"type": "equation",
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"img_path": "images/b5c27ea925b0d009a8f43c195c902d40ad96012b0d9951f391fc954740b4293c.jpg",
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"text": "$$\n\\begin{array} { r l r } { { p ^ { \\mathrm { c o r e f } } ( v _ { i } | h _ { i - 1 } , h ^ { e } ) = \\mathrm { A T T N } ( h ^ { e } , h _ { i - 1 } ) , } } \\\\ & { } & { d _ { i } = \\sum _ { v _ { i } } p ( v _ { i } ) h _ { v _ { i } } ^ { e } } \\\\ & { } & { p ( z _ { i } | h _ { i - 1 } ) = \\mathrm { s i g m o i d } ( W [ d _ { i } , h _ { i - 1 } ] ) , } \\end{array}\n$$",
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"text_format": "latex",
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{
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| 858 |
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"type": "text",
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| 859 |
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"text": "where $z _ { i }$ denotes whether the next word is an entity and if yes $v _ { i }$ denotes which entity the next word corefers to. If the next word is an entity mention, then $p ( x _ { i } | v _ { i } , h _ { i - 1 } , h ^ { e } ) =$ softmax $( W _ { 1 } \\operatorname { t a n h } ( W _ { 2 } [ h _ { v _ { i } } ^ { e } , h _ { i - 1 } ] ) )$ else $p ( x _ { i } | h _ { i - 1 } ) = \\mathrm { s o f t m a x } ( W _ { 1 } h _ { i - 1 } )$ , ",
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"img_path": "images/e5d980b4e67b4495600016bae44aca7cb6065cf154c0da9ccd8f88373126db37.jpg",
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"text": "$$\np ( x _ { i } | x _ { < i } ) = \\left\\{ \\begin{array} { l l } { p ( x _ { i } | h _ { i - 1 } ) p ( z _ { i } | h _ { i - 1 } , h ^ { e } ) } & { \\quad \\mathrm { i f } \\quad z _ { i } = 0 . } \\\\ { p ( x _ { i } | v _ { i } , h _ { i - 1 } , h ^ { e } ) p ^ { \\mathrm { c o r e f } } ( v _ { i } | h _ { i - 1 } , h ^ { e } ) p ( z _ { i } | h _ { i - 1 } , h ^ { e } ) } & { \\quad \\mathrm { i f } \\quad z _ { i } = 1 . } \\end{array} \\right.\n$$",
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"text": "Entity state update: We update the entity state $h ^ { e }$ at each time step. In the beginning, $h ^ { e } = \\{ h _ { 0 } ^ { e } \\}$ , $h _ { 0 } ^ { e }$ denotes the state of an virtual empty entity and is a learnable variable. If $z _ { i } = 1$ and $v _ { i } = 0$ , then it indicates the next word is a new entity mention, then in the next step, we append $h _ { i }$ to $h ^ { e }$ , i.e., $h ^ { e } = \\{ h ^ { e } , h _ { i } \\}$ , if $e _ { i } > 0$ , then we update the corresponding entity state with the new hidden state, $h ^ { e } [ v _ { i } ] = h _ { i }$ . Another way to update the entity state is to use one LSTM to encode the mention states and get the new entity state. Here we use the latest entity mention state as the new entity state for simplicity. The detailed update process is shown in Figure 5. ",
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"type": "text",
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"text": "3 EXPERIMENTS ",
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"type": "text",
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"text": "4 DATA SETS AND PREPROCESSING ",
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"text": "Dialogue: We use the DSTC2 data set. We only extracted the dialogue transcript from data set. There are about 3,200 dialogues in total. Since this is a small data set, we use 5-fold cross validation and report the average result over the 5 partitions. There may be multiple tokens in each table cell, for example in Table.2, the name, address, post code and phone number have multiple tokens, we replace them with one special token. For the name, address, post code and phone number of the $j$ -th row, we replace the tokens in each cell with NAME $j$ , ADDR $j$ , POSTCODE $j$ , PHONE $j$ . If a table cell is empty, we replace it with an empty token EMPTY. We do a string match in the transcript and replace the corresponding tokens in transcripts from the table with the special tokens. ",
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"text": "Each dialogue on average has 8 turns (16 sentences). We use a vocabulary size of 900, including about 400 table tokens and 500 words. ",
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"type": "text",
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"text": "Recipes: We crawl all recipes from www.allrecipes.com. There are about 31, 000 recipes in total, and every recipe has a ingredient list and a corresponding recipe. We exclude the recipes that have less than 10 tokens or more than 500 tokens, those recipes take about $0 . 1 \\%$ of all data set. On average each recipe has 118 tokens and 9 ingredients. We random shuffle the whole data set and take $80 \\%$ as training and $10 \\%$ for validation and test. We use a vocabulary size of 10,000 in the model. ",
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"text": "Coref LM: We use the Xinhua News data set from Gigaword Fifth Edition and sample 100,000 documents from it that has length in range from 100 to 500. Each document has on average 234 tokens, so there are 23 million tokens in total. We use a tool to annotate all the entity mentions and use the annotation in the training. We take $80 \\%$ as training and $10 \\%$ as validation and test respectively. We ignore the entities that have only one mention and for the mentions that have multiple tokens, we take the token that is most frequent in the all the mentions for this entity. After the preprocessing, tokens that are entity mentions take about $10 \\%$ of all tokens. We use a vocabulary size of 50,000 in the model. ",
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"text": "4.1 MODEL TRAINING AND EVALUATION ",
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"text": "We train all models with simple stochastic gradient descent with clipping. We use a one-layer LSTM for all RNN components. Hyper-parameters are selected using grid search based on the validation set. We use dropout after the input embedding and LSTM output. The learning rate is selected from [0.1, 0.2, 0.5, 1], maximum gradient norm is selected from [1, 2, 5, 10] and drop ratio is selected from [0.2, 0.3, 0.5]. The batch size and LSTM dimension size is slightly different for different tasks so as to make the model fit into memory. The number of epochs to train are different for each task and we drop the learning rate after reaching a given number of epochs. We report the per-word perplexity for all tasks, specifically, we report the perplexity of all words, words that can be generated from reference and non-reference words. For recipe generation, we also generate the recipe using beam size of 10 and evaluate the generated recipe with BLEU. ",
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{
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"type": "table",
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"img_path": "images/d3108c6d0c18ea8ba368c798cbd6f63b81f84be385fb73c6403abc76351dad2e.jpg",
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"table_caption": [],
|
| 987 |
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"table_footnote": [],
|
| 988 |
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"table_body": "<table><tr><td>model</td><td>all</td><td>table</td><td>table oov</td><td>word</td></tr><tr><td>seq2seq</td><td>1.35±0.01</td><td>4.98±0.38</td><td>1.99E7±7.75E6</td><td>1.23±0.01</td></tr><tr><td>table attn</td><td>1.37±0.01</td><td>5.09±0.64</td><td>7.91E7±1.39E8</td><td>1.24±0.01</td></tr><tr><td>table pointer</td><td>1.33±0.01</td><td>3.99±0.36</td><td>1360 ± 2600</td><td>1.23±0.01</td></tr><tr><td>table latent</td><td>1.36±0.01</td><td>4.99±0.20</td><td>3.78E7±6.08E7</td><td>1.24±0.01</td></tr><tr><td colspan=\"5\">+ sentence attn</td></tr><tr><td>seq2seq</td><td>1.28±0.01</td><td>3.31±0.21</td><td>2.83E9±4.69E9</td><td>1.19±0.01</td></tr><tr><td>table attn</td><td>1.28±0.01</td><td>3.17±0.21</td><td>1.67E7±9.5E6</td><td>1.20±0.01</td></tr><tr><td>table pointer</td><td>1.27±0.01</td><td>2.99±0.19</td><td>82.86±110</td><td>1.20±0.01</td></tr><tr><td>table latent</td><td>1.28±0.01</td><td>3.26±0.25</td><td>1.27E7±1.41E7</td><td>1.20±0.01</td></tr></table>",
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"text": "Table 4: Dialogue perplexity results. (All means all tokens, table means tokens from table, table oov denotes table tokens that does not appear in the training set, word means non-table tokens). sentence attn denotes we use attention mechanism over tokens from past turn. Table pointer and table latent differs in that table pointer, we provide supervised signal on when to generate a table token, while in table latent it is a latent decision. ",
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| 1009 |
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|
| 1010 |
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"img_path": "images/4bf9239ef5f04ab5432eae55b9a70f34a73dae3493ed69a908a9d5fd13f4c2f1.jpg",
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"table_caption": [],
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"table_footnote": [],
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| 1013 |
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"table_body": "<table><tr><td rowspan=\"3\">model</td><td colspan=\"4\">val</td><td colspan=\"4\">test</td></tr><tr><td colspan=\"3\">ppl</td><td rowspan=\"2\">BLEU</td><td colspan=\"3\">ppl</td><td rowspan=\"2\">BLEU</td></tr><tr><td>all</td><td>ing</td><td>word</td><td>all</td><td>ing</td><td>word</td></tr><tr><td>seq2seq</td><td>5.60</td><td>11.26</td><td>5.00</td><td>14.07</td><td>5.52</td><td>11.26</td><td>4.91</td><td>14.39</td></tr><tr><td>attn</td><td>5.25</td><td>6.86</td><td>5.03</td><td>14.84</td><td>5.19</td><td>6.92</td><td>4.95</td><td>15.15</td></tr><tr><td>pointer</td><td>5.15</td><td>5.86</td><td>5.04</td><td>15.06</td><td>5.11</td><td>6.04</td><td>4.98</td><td>15.29</td></tr><tr><td>latent</td><td>5.02</td><td>5.10</td><td>5.01</td><td>14.87</td><td>4.97</td><td>5.19</td><td>4.94</td><td>15.41</td></tr></table>",
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| 1023 |
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|
| 1024 |
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"text": "Table 5: Recipe result, evaluated in perplexity and BLEU score. ing denotes tokens from recipe that appear in ingredients. ",
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"img_path": "images/ed337d504821f92e0a5582716c9ae60118e31da5536dda3a598f2c873c9b12f5.jpg",
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| 1037 |
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| 1038 |
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"table_body": "<table><tr><td rowspan=\"2\">model</td><td colspan=\"3\">val</td><td colspan=\"3\">test</td></tr><tr><td>all</td><td>entity</td><td>word</td><td>all</td><td>entity</td><td>word</td></tr><tr><td>lm</td><td>33.08</td><td>44.52</td><td>32.04</td><td>33.08</td><td>43.86</td><td>32.10</td></tr><tr><td>pointer</td><td>32.57</td><td>32.07</td><td>32.62</td><td>32.62</td><td>32.07</td><td>32.69</td></tr><tr><td>pointer +init</td><td>30.43</td><td>28.56</td><td>30.63</td><td>30.42</td><td>28.56</td><td>30.66</td></tr></table>",
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"type": "text",
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| 1049 |
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"text": "Table 6: Coreference based LM. pointer $^ +$ init means we initialize the model with the LM weights. ",
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| 1050 |
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"type": "text",
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| 1060 |
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"text": "4.2 RESULTS AND ANALYSIS ",
|
| 1061 |
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| 1072 |
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"text": "The results for dialogue, recipe generation and coref language model are shown in Table 4, 5 and 6 respectively. We can see from Table 4 that models that condition on table performs better in predicting table tokens in general. Table pointer has the lowest perplexity for token in the table. Since the table token appears rarely in the dialogue, the overall perplexity does not differ much and the non-table tokens perplexity are similar. With attention mechanism over the table, the perplexity of table token improves over basic seq2seq model, but not as good as directly pointing to cells in the table. As expected, using sentence attention improves significantly over models without sentence attention. Surprisingly, table latent performs much worse than table pointer. We also measure the perplexity of table tokens that appear only in test set. For models other than table pointer, because the tokens never appear in training set, the perplexity is quite high, while table pointer can predict these tokens much more accurately. The recipe results in Table 5 in general follows that findings from the dialogue. But the latent model performs better than pointer model since that tokens in ingredients that match with recipe does not necessarily come from the ingredients. Imposing a supervised signal will give wrong information to the model and hence make the result worse. Hence with latent decision, the model learns to when to copy and when to generate it from the vocabulary. The coref LM results are shown in Table 6. We find that coref based LM performs much better on the entities perplexities, but however is a little bit worse than for non-entity words. We found it is an optimization problem and perhaps the model is stuck in local optimum. So we initialize the pointer model with the weights learned from LM, the pointer model performs better than LM both for entity perplexity and non-entity words perplexity. ",
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| 1073 |
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"page_idx": 8
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{
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"type": "text",
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"text": "5 RELATED WORK ",
|
| 1084 |
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"text_level": 1,
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"bbox": [
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},
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| 1093 |
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{
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"type": "text",
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"text": "Recently, there has been great progresses in modeling languages based on neural network, including language modeling (Mikolov et al., 2010; Jozefowicz et al., 2016), machine translation (Sutskever et al., 2014; Bahdanau et al., 2014), question answering (Hermann et al., 2015) etc. Based on the success of seq2seq models, neural networks are applied in modeling chit-chat dialogue (Li et al., 2016; Vinyals & Le, 2015; Sordoni et al., 2015; Serban et al., 2016; Shang et al., 2015) and task oriented dialogue (Wen et al., 2015; Bordes & Weston, 2016; Williams & Zweig, 2016; Wen et al., 2016). Most of the chit-chat neural dialogue models are simply applying the seq2seq models. For the task oriented dialogues, most of them embed the seq2seq model in traditional dialogue systems, in which the table query part is not differentiable. while our model queries the database directly. Recipe generation was proposed in (Kiddon et al., 2016). Their model extents previous work on attention models (Allamanis et al., 2016) to checklists, whereas our work models explicit references to those checklists. Context dependent language models (Mikolov et al., 2010; Ji et al., 2015; Wang & Cho, 2015) are proposed to capture long term dependency of text. There are also lots of works on coreference resolution (Haghighi & Klein, 2010; Wiseman et al., 2016). We are the first to combine coreference with language modeling, to the best of our knowledge. Much effort has been invested in embedding a copying mechanism for neural models (Gulc¸ehre et al. ¨ , 2016; Gu et al., 2016; Ling et al., 2016). In general, a gating mechanism is employed to combine the softmax over observed words and a pointer network (Vinyals et al., 2015). These gates can be trained either by marginalizing over both outcomes, or using heuristics (e.g. copy low frequency words). Our models are similar to models proposed in (Ahn et al., 2016; Merity et al., 2016), where the generation of each word can be conditioned on a particular entry in knowledge lists and previous words. In our work, we describe a model with broader applications, allowing us to condition, on databases, lists and dynamic lists. ",
|
| 1096 |
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{
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"type": "text",
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"text": "6 CONCLUSION ",
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"text_level": 1,
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"bbox": [
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"text": "We introduce reference-aware language models which explicitly model the decision of from where to generate the token at each step. Our model can also learns the decision by treating it as a latent variable. We demonstrate on three tasks, table based dialogue modeling, recipe generation and coref based LM, that our model performs better than attention based model, which does not incorporate this decision explicitly. There are several directions to explore further based on our framework. The current evaluation method is based on perplexity and BLEU. In task oriented dialogues, we can also try human evaluation to see if the model can reply users’ query accurately. It is also interesting to use reinforcement learning to learn the actions in each step. ",
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"text": "Tsung-Hsien Wen, Milica Gasic, Nikola Mrksic, Lina M Rojas-Barahona, Pei-Hao Su, Stefan Ultes, David Vandyke, and Steve Young. A network-based end-to-end trainable task-oriented dialogue system. arXiv preprint arXiv:1604.04562, 2016. ",
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"type": "text",
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"text": "Jason D Williams and Geoffrey Zweig. End-to-end lstm-based dialog control optimized with supervised and reinforcement learning. arXiv preprint arXiv:1606.01269, 2016. ",
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"type": "text",
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"text": "Sam Wiseman, Alexander M Rush, and Stuart M Shieber. Learning global features for coreference resolution. arXiv preprint arXiv:1604.03035, 2016. ",
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"type": "text",
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"text": "Steve Young, Milica Gasiˇ c, Blaise Thomson, and Jason D Williams. Pomdp-based statistical spoken ´ dialog systems: A review. Proceedings of the IEEE, 101(5):1160–1179, 2013. ",
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"type": "text",
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"text": "A VISUALIZATION ",
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"text": "M: Hello , welcome to the Cambridge restaurant system ? You can ask for restaurants by area , price range or food type . How may I help you ? \nU: and expensive restaurant that serves tuscan food \nM: Sorry there is no tuscan restaurant in the expensive price range \nU: french food \nM: cote is a great restaurant serving french food and it is in the expensive price range \nU: may i have the phone number \nM: The phone number of cote is 01223 311053 . \nU: thank you good bye \nM: Hello , welcome to the Cambridge restaurant system ? You can ask for restaurants by area , price range or food type . How may I help you ? \nU: cheap restaurant \nM: What kind of food would you like ? \nU: in the west part of town \nM: la margherita is a nice restaurant in the west of town in the cheap price range \nU: address \nM: Sure , la margherita is on 15 Magdalene Street City Centre \nU: thank you \nM: la margherita is a nice restaurant in the west of town in the cheap price range \nU: good bye ",
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"table_body": "<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=1>PRICE RANGE</td><td rowspan=1 colspan=1>FOOD</td><td rowspan=1 colspan=1>AREA</td><td rowspan=1 colspan=1>ADDRESS</td><td rowspan=1 colspan=1>POST CODE</td><td rowspan=1 colspan=1>PHONE</td></tr><tr><td rowspan=1 colspan=1>charlie chan</td><td rowspan=1 colspan=1>cheap</td><td rowspan=1 colspan=1>chinese</td><td rowspan=1 colspan=1>east</td><td rowspan=1 colspan=1>Regent Street City Cen-tre</td><td rowspan=1 colspan=1>C.B 2, 1 D.B</td><td rowspan=1 colspan=1>01223 361763</td></tr><tr><td rowspan=1 colspan=1>chiquito restau-rant bar</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>mexican</td><td rowspan=1 colspan=1>south</td><td rowspan=1 colspan=1>2G Cambridge LeisurePark Cherry HintonRoad Cherry Hinton</td><td rowspan=1 colspan=1>C.B 1,7D.Y</td><td rowspan=1 colspan=1>01223 400170</td></tr><tr><td rowspan=1 colspan=1>city stop</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>food</td><td rowspan=1 colspan=1>north</td><td rowspan=1 colspan=1>Cambridge City Foot-ball Club Milton RoadChesterton</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 363270</td></tr><tr><td rowspan=1 colspan=1>clowns cafe</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>italian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>C.B 1,1 L.N</td><td rowspan=1 colspan=1>01223 355711</td></tr><tr><td rowspan=1 colspan=1>cocum</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>west</td><td rowspan=1 colspan=1>71 CastleStreet CityCentre</td><td rowspan=1 colspan=1>C.B 3,0 A.H</td><td rowspan=1 colspan=1>01223 366668</td></tr><tr><td rowspan=1 colspan=1>cote</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>french</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>Bridge Street City Cen-tre</td><td rowspan=1 colspan=1>C.B 2, 1U.F</td><td rowspan=1 colspan=1>01223 311053</td></tr><tr><td rowspan=1 colspan=1> curry garden</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>106 Regent Street CityCentre</td><td rowspan=1 colspan=1>EMPTY</td><td rowspan=1 colspan=1>01223 302330</td></tr><tr><td rowspan=1 colspan=1>curry king</td><td rowspan=1 colspan=1>expensive</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>centre</td><td rowspan=1 colspan=1>5Jordans Yard BridgeStreet City Centre</td><td rowspan=1 colspan=1>C.B 1,2 B.D</td><td rowspan=1 colspan=1>01223 324351</td></tr><tr><td rowspan=1 colspan=1>curry prince</td><td rowspan=1 colspan=1>moderate</td><td rowspan=1 colspan=1>indian</td><td rowspan=1 colspan=1>east</td><td rowspan=1 colspan=1>451 Newmarket RoadFen Ditton</td><td rowspan=1 colspan=1>C.B 5, 8 J.J</td><td rowspan=1 colspan=1>01223 566388</td></tr></table>",
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"text": "(c) Attention heap map: Sure , la margherita is on 15 Magdalene Street City Centre. ",
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| 1567 |
+
844
|
| 1568 |
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|
| 1569 |
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"page_idx": 12
|
| 1570 |
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},
|
| 1571 |
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{
|
| 1572 |
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"type": "image",
|
| 1573 |
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"img_path": "images/5c040a0339d10ef5db2a8c0736a89dc5f36207d727dc95cefe05eff4a67e065b.jpg",
|
| 1574 |
+
"image_caption": [
|
| 1575 |
+
"Figure 6: Recipe heat map example 1. The ingredient tokens appear on the left while the recipe tokens appear on the top. The first row is the $p \\big ( \\bar { z } _ { v } | s _ { v } \\big )$ . "
|
| 1576 |
+
],
|
| 1577 |
+
"image_footnote": [],
|
| 1578 |
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"bbox": [
|
| 1579 |
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179,
|
| 1580 |
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|
| 1581 |
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820,
|
| 1582 |
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763
|
| 1583 |
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|
| 1584 |
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"page_idx": 13
|
| 1585 |
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|
| 1586 |
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{
|
| 1587 |
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"type": "image",
|
| 1588 |
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"img_path": "images/540e5f6a28797252925d3445abfb72ca056f1c0a464097561f7c4b5ff011b8df.jpg",
|
| 1589 |
+
"image_caption": [
|
| 1590 |
+
"Figure 7: Recipe heat map example 2. "
|
| 1591 |
+
],
|
| 1592 |
+
"image_footnote": [],
|
| 1593 |
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"bbox": [
|
| 1594 |
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181,
|
| 1595 |
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| 1596 |
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823,
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| 1597 |
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|
| 1598 |
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|
| 1599 |
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"page_idx": 14
|
| 1600 |
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}
|
| 1601 |
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]
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