evalstate/spherical-equivariance-classifiers

Spherical-equivariance classifiers (spectral conv vs SoLA)

Tiny spherical-MNIST classifiers from the Spherical Equivariance Benchmark, built to compare classical zonal spectral spherical convolution (Cohen et al., ICML 2018, arXiv:1711.06721) with SoLA local SO(3)-equivariant attention (Sekikawa et al., ICML 2026 #1440, OpenReview S1StjDehJS).

These are intentionally tiny (≤2k parameters) and trained at small scale to fit a local-compute budget; they are not paper-scale accuracy models. Their job is to demonstrate the equivariance contrast on a real spherical image task.

Architectures (rank-2 icosphere, 162 vertices; dim=16; lmax=6; 9 classes)

• spectral.pt — two zonal SpectralConvLayers (exact complex SHT via least-squares pseudo-inverse + per-degree scalar filter) → mean-pool → linear head. 2057 params.
• sola.pt — two SoLALayers (score(i,j)=<q_i,k_j>·(b+w·u_i·u_j) over mesh 2-ring neighborhoods) → mean-pool → linear head. 973 params.
• mlp.pt — per-vertex MLP baseline (non-equivariant). 1001 params.
• mlp_aug.pt — same MLP, trained with random SO(3) augmentation (empirically-learned equivariance). 1001 params.

Checkpoints store learnable parameters only (the complex SHT basis buffers and the icosphere geometry are deterministic and rebuilt by make_model(...) in train.py). Load with strict=False.

Results (25-40 epochs, CPU)

Model	canonical acc	rotated acc (mean of 3)	params	latency (ms, batch-1 CPU)
spectral conv	45.2%	40.7%	2057	2.31
SoLA	17.0%	17.7%	973	2.35
MLP (no aug)	21.8%	22.1%	1001	0.31
MLP (+ rot aug)	24.9%	25.1%	1001	1.84

All models underfit at this scale; the point is that the equivariant models retain rotated ≈ canonical accuracy by construction (e.g. spectral 40.7 vs 45.2), even when underfit. Paper-scale rotation robustness (SoLA 71.8 vs a position-embedding baseline 13.5 under SO(3) test rotation) is reproduced in the SoLA reproduction logbook.

Checkpoint + code locations (canonical, in the research bucket)

• checkpoints: hf://buckets/evalstate/research-agent/26-08-21-classical-vs-modern-spherical-6ab9/scratch/research/data/checkpoints/ (HTTPS)
• model/training code: train.py under .../scratch/research/code/
• full comparative report: output/report.md