Daily Papers

For image generation with diffusion models (DMs), a negative prompt n can be used to complement the text prompt p, helping define properties not desired in the synthesized image. While this improves prompt adherence and image quality, finding good negative prompts is challenging. We argue that this is due to a semantic gap between humans and DMs, which makes good negative prompts for DMs appear unintuitive to humans. To bridge this gap, we propose a new diffusion-negative prompting (DNP) strategy. DNP is based on a new procedure to sample images that are least compliant with p under the distribution of the DM, denoted as diffusion-negative sampling (DNS). Given p, one such image is sampled, which is then translated into natural language by the user or a captioning model, to produce the negative prompt n*. The pair (p, n*) is finally used to prompt the DM. DNS is straightforward to implement and requires no training. Experiments and human evaluations show that DNP performs well both quantitatively and qualitatively and can be easily combined with several DM variants.

With growing attention to tabular data these days, the attempt to apply a synthetic table to various tasks has been expanded toward various scenarios. Owing to the recent advances in generative modeling, fake data generated by tabular data synthesis models become sophisticated and realistic. However, there still exists a difficulty in modeling discrete variables (columns) of tabular data. In this work, we propose to process continuous and discrete variables separately (but being conditioned on each other) by two diffusion models. The two diffusion models are co-evolved during training by reading conditions from each other. In order to further bind the diffusion models, moreover, we introduce a contrastive learning method with a negative sampling method. In our experiments with 11 real-world tabular datasets and 8 baseline methods, we prove the efficacy of the proposed method, called CoDi.

Text-to-image diffusion models have shown great success in generating high-quality text-guided images. Yet, these models may still fail to semantically align generated images with the provided text prompts, leading to problems like incorrect attribute binding and/or catastrophic object neglect. Given the pervasive object-oriented structure underlying text prompts, we introduce a novel object-conditioned Energy-Based Attention Map Alignment (EBAMA) method to address the aforementioned problems. We show that an object-centric attribute binding loss naturally emerges by approximately maximizing the log-likelihood of a z-parameterized energy-based model with the help of the negative sampling technique. We further propose an object-centric intensity regularizer to prevent excessive shifts of objects attention towards their attributes. Extensive qualitative and quantitative experiments, including human evaluation, on several challenging benchmarks demonstrate the superior performance of our method over previous strong counterparts. With better aligned attention maps, our approach shows great promise in further enhancing the text-controlled image editing ability of diffusion models.

Negative guidance -- explicitly suppressing unwanted attributes -- remains a fundamental challenge in diffusion models, particularly in few-step sampling regimes. While Classifier-Free Guidance (CFG) works well in standard settings, it fails under aggressive sampling step compression due to divergent predictions between positive and negative branches. We present Normalized Attention Guidance (NAG), an efficient, training-free mechanism that applies extrapolation in attention space with L1-based normalization and refinement. NAG restores effective negative guidance where CFG collapses while maintaining fidelity. Unlike existing approaches, NAG generalizes across architectures (UNet, DiT), sampling regimes (few-step, multi-step), and modalities (image, video), functioning as a universal plug-in with minimal computational overhead. Through extensive experimentation, we demonstrate consistent improvements in text alignment (CLIP Score), fidelity (FID, PFID), and human-perceived quality (ImageReward). Our ablation studies validate each design component, while user studies confirm significant preference for NAG-guided outputs. As a model-agnostic inference-time approach requiring no retraining, NAG provides effortless negative guidance for all modern diffusion frameworks -- pseudocode in the Appendix!

Training-free diffusion-based methods have achieved remarkable success in style transfer, eliminating the need for extensive training or fine-tuning. However, due to the lack of targeted training for style information extraction and constraints on the content image layout, training-free methods often suffer from layout changes of original content and content leakage from style images. Through a series of experiments, we discovered that an effective startpoint in the sampling stage significantly enhances the style transfer process. Based on this discovery, we propose StyleSSP, which focuses on obtaining a better startpoint to address layout changes of original content and content leakage from style image. StyleSSP comprises two key components: (1) Frequency Manipulation: To improve content preservation, we reduce the low-frequency components of the DDIM latent, allowing the sampling stage to pay more attention to the layout of content images; and (2) Negative Guidance via Inversion: To mitigate the content leakage from style image, we employ negative guidance in the inversion stage to ensure that the startpoint of the sampling stage is distanced from the content of style image. Experiments show that StyleSSP surpasses previous training-free style transfer baselines, particularly in preserving original content and minimizing the content leakage from style image.

In image editing employing diffusion models, it is crucial to preserve the reconstruction quality of the original image while changing its style. Although existing methods ensure reconstruction quality through optimization, a drawback of these is the significant amount of time required for optimization. In this paper, we propose negative-prompt inversion, a method capable of achieving equivalent reconstruction solely through forward propagation without optimization, thereby enabling much faster editing processes. We experimentally demonstrate that the reconstruction quality of our method is comparable to that of existing methods, allowing for inversion at a resolution of 512 pixels and with 50 sampling steps within approximately 5 seconds, which is more than 30 times faster than null-text inversion. Reduction of the computation time by the proposed method further allows us to use a larger number of sampling steps in diffusion models to improve the reconstruction quality with a moderate increase in computation time.

We present a novel approach for generating minority samples that live on low-density regions of a data manifold. Our framework is built upon diffusion models, leveraging the principle of guided sampling that incorporates an arbitrary energy-based guidance during inference time. The key defining feature of our sampler lies in its self-contained nature, \ie, implementable solely with a pretrained model. This distinguishes our sampler from existing techniques that require expensive additional components (like external classifiers) for minority generation. Specifically, we first estimate the likelihood of features within an intermediate latent sample by evaluating a reconstruction loss w.r.t. its posterior mean. The generation then proceeds with the minimization of the estimated likelihood, thereby encouraging the emergence of minority features in the latent samples of subsequent timesteps. To further improve the performance of our sampler, we provide several time-scheduling techniques that properly manage the influence of guidance over inference steps. Experiments on benchmark real datasets demonstrate that our approach can greatly improve the capability of creating realistic low-likelihood minority instances over the existing techniques without the reliance on costly additional elements. Code is available at https://github.com/soobin-um/sg-minority.

We explore the problem of generating minority samples using diffusion models. The minority samples are instances that lie on low-density regions of a data manifold. Generating a sufficient number of such minority instances is important, since they often contain some unique attributes of the data. However, the conventional generation process of the diffusion models mostly yields majority samples (that lie on high-density regions of the manifold) due to their high likelihoods, making themselves ineffective and time-consuming for the minority generating task. In this work, we present a novel framework that can make the generation process of the diffusion models focus on the minority samples. We first highlight that Tweedie's denoising formula yields favorable results for majority samples. The observation motivates us to introduce a metric that describes the uniqueness of a given sample. To address the inherent preference of the diffusion models w.r.t. the majority samples, we further develop minority guidance, a sampling technique that can guide the generation process toward regions with desired likelihood levels. Experiments on benchmark real datasets demonstrate that our minority guidance can greatly improve the capability of generating high-quality minority samples over existing generative samplers. We showcase that the performance benefit of our framework persists even in demanding real-world scenarios such as medical imaging, further underscoring the practical significance of our work. Code is available at https://github.com/soobin-um/minority-guidance.

Diffusion-based generative models have achieved remarkable success in various domains. It trains a shared model on denoising tasks that encompass different noise levels simultaneously, representing a form of multi-task learning (MTL). However, analyzing and improving diffusion models from an MTL perspective remains under-explored. In particular, MTL can sometimes lead to the well-known phenomenon of negative transfer, which results in the performance degradation of certain tasks due to conflicts between tasks. In this paper, we first aim to analyze diffusion training from an MTL standpoint, presenting two key observations: (O1) the task affinity between denoising tasks diminishes as the gap between noise levels widens, and (O2) negative transfer can arise even in diffusion training. Building upon these observations, we aim to enhance diffusion training by mitigating negative transfer. To achieve this, we propose leveraging existing MTL methods, but the presence of a huge number of denoising tasks makes this computationally expensive to calculate the necessary per-task loss or gradient. To address this challenge, we propose clustering the denoising tasks into small task clusters and applying MTL methods to them. Specifically, based on (O2), we employ interval clustering to enforce temporal proximity among denoising tasks within clusters. We show that interval clustering can be solved using dynamic programming, utilizing signal-to-noise ratio, timestep, and task affinity for clustering objectives. Through this, our approach addresses the issue of negative transfer in diffusion models by allowing for efficient computation of MTL methods. We validate the proposed clustering and its integration with MTL methods through various experiments, demonstrating improved sample quality of diffusion models. Our project page is available at https://gohyojun15.github.io/ANT_diffusion/{url}.

Diffusion models, as a kind of powerful generative model, have given impressive results on image super-resolution (SR) tasks. However, due to the randomness introduced in the reverse process of diffusion models, the performances of diffusion-based SR models are fluctuating at every time of sampling, especially for samplers with few resampled steps. This inherent randomness of diffusion models results in ineffectiveness and instability, making it challenging for users to guarantee the quality of SR results. However, our work takes this randomness as an opportunity: fully analyzing and leveraging it leads to the construction of an effective plug-and-play sampling method that owns the potential to benefit a series of diffusion-based SR methods. More in detail, we propose to steadily sample high-quality SR images from pre-trained diffusion-based SR models by solving diffusion ordinary differential equations (diffusion ODEs) with optimal boundary conditions (BCs) and analyze the characteristics between the choices of BCs and their corresponding SR results. Our analysis shows the route to obtain an approximately optimal BC via an efficient exploration in the whole space. The quality of SR results sampled by the proposed method with fewer steps outperforms the quality of results sampled by current methods with randomness from the same pre-trained diffusion-based SR model, which means that our sampling method "boosts" current diffusion-based SR models without any additional training.

We introduce ensembles within score-based sampling methods to develop gradient-free approximate sampling techniques that leverage the collective dynamics of particle ensembles to compute approximate reverse diffusion drifts. We introduce the underlying methodology, emphasizing its relationship with generative diffusion models and the previously introduced F\"ollmer sampler. We demonstrate the efficacy of ensemble strategies through various examples, ranging from low- to medium-dimensionality sampling problems, including multi-modal and highly non-Gaussian probability distributions, and provide comparisons to traditional methods like NUTS. Our findings highlight the potential of ensemble strategies for modeling complex probability distributions in situations where gradients are unavailable. Finally, we showcase its application in the context of Bayesian inversion problems within the geophysical sciences.

Denoising diffusion models are a powerful type of generative models used to capture complex distributions of real-world signals. However, their applicability is limited to scenarios where training samples are readily available, which is not always the case in real-world applications. For example, in inverse graphics, the goal is to generate samples from a distribution of 3D scenes that align with a given image, but ground-truth 3D scenes are unavailable and only 2D images are accessible. To address this limitation, we propose a novel class of denoising diffusion probabilistic models that learn to sample from distributions of signals that are never directly observed. Instead, these signals are measured indirectly through a known differentiable forward model, which produces partial observations of the unknown signal. Our approach involves integrating the forward model directly into the denoising process. This integration effectively connects the generative modeling of observations with the generative modeling of the underlying signals, allowing for end-to-end training of a conditional generative model over signals. During inference, our approach enables sampling from the distribution of underlying signals that are consistent with a given partial observation. We demonstrate the effectiveness of our method on three challenging computer vision tasks. For instance, in the context of inverse graphics, our model enables direct sampling from the distribution of 3D scenes that align with a single 2D input image.

Denoising diffusion probabilistic models (DDPM) are a class of generative models which have recently been shown to produce excellent samples. We show that with a few simple modifications, DDPMs can also achieve competitive log-likelihoods while maintaining high sample quality. Additionally, we find that learning variances of the reverse diffusion process allows sampling with an order of magnitude fewer forward passes with a negligible difference in sample quality, which is important for the practical deployment of these models. We additionally use precision and recall to compare how well DDPMs and GANs cover the target distribution. Finally, we show that the sample quality and likelihood of these models scale smoothly with model capacity and training compute, making them easily scalable. We release our code at https://github.com/openai/improved-diffusion

Diffusion models have been recently studied as powerful generative inverse problem solvers, owing to their high quality reconstructions and the ease of combining existing iterative solvers. However, most works focus on solving simple linear inverse problems in noiseless settings, which significantly under-represents the complexity of real-world problems. In this work, we extend diffusion solvers to efficiently handle general noisy (non)linear inverse problems via approximation of the posterior sampling. Interestingly, the resulting posterior sampling scheme is a blended version of diffusion sampling with the manifold constrained gradient without a strict measurement consistency projection step, yielding a more desirable generative path in noisy settings compared to the previous studies. Our method demonstrates that diffusion models can incorporate various measurement noise statistics such as Gaussian and Poisson, and also efficiently handle noisy nonlinear inverse problems such as Fourier phase retrieval and non-uniform deblurring. Code available at https://github.com/DPS2022/diffusion-posterior-sampling

Generative models have the potential to accelerate key steps in the discovery of novel molecular therapeutics and materials. Diffusion models have recently emerged as a powerful approach, excelling at unconditional sample generation and, with data-driven guidance, conditional generation within their training domain. Reliably sampling from high-value regions beyond the training data, however, remains an open challenge -- with current methods predominantly focusing on modifying the diffusion process itself. In this paper, we develop context-guided diffusion (CGD), a simple plug-and-play method that leverages unlabeled data and smoothness constraints to improve the out-of-distribution generalization of guided diffusion models. We demonstrate that this approach leads to substantial performance gains across various settings, including continuous, discrete, and graph-structured diffusion processes with applications across drug discovery, materials science, and protein design.

Diffusion models recently proved to be remarkable priors for Bayesian inverse problems. However, training these models typically requires access to large amounts of clean data, which could prove difficult in some settings. In this work, we present a novel method based on the expectation-maximization algorithm for training diffusion models from incomplete and noisy observations only. Unlike previous works, our method leads to proper diffusion models, which is crucial for downstream tasks. As part of our method, we propose and motivate an improved posterior sampling scheme for unconditional diffusion models. We present empirical evidence supporting the effectiveness of our method.

Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step followed by a projection-based measurement consistency step, often produce suboptimal results. By studying the generative sampling path, here we show that current solvers throw the sample path off the data manifold, and hence the error accumulates. To address this, we propose an additional correction term inspired by the manifold constraint, which can be used synergistically with the previous solvers to make the iterations close to the manifold. The proposed manifold constraint is straightforward to implement within a few lines of code, yet boosts the performance by a surprisingly large margin. With extensive experiments, we show that our method is superior to the previous methods both theoretically and empirically, producing promising results in many applications such as image inpainting, colorization, and sparse-view computed tomography. Code available https://github.com/HJ-harry/MCG_diffusion

While conditional diffusion models are known to have good coverage of the data distribution, they still face limitations in output diversity, particularly when sampled with a high classifier-free guidance scale for optimal image quality or when trained on small datasets. We attribute this problem to the role of the conditioning signal in inference and offer an improved sampling strategy for diffusion models that can increase generation diversity, especially at high guidance scales, with minimal loss of sample quality. Our sampling strategy anneals the conditioning signal by adding scheduled, monotonically decreasing Gaussian noise to the conditioning vector during inference to balance diversity and condition alignment. Our Condition-Annealed Diffusion Sampler (CADS) can be used with any pretrained model and sampling algorithm, and we show that it boosts the diversity of diffusion models in various conditional generation tasks. Further, using an existing pretrained diffusion model, CADS achieves a new state-of-the-art FID of 1.70 and 2.31 for class-conditional ImageNet generation at 256times256 and 512times512 respectively.

Generative diffusion models have emerged as a powerful tool for high-quality image synthesis, yet their iterative nature demands significant computational resources. This paper proposes an efficient time step sampling method based on an image spectral analysis of the diffusion process, aimed at optimizing the denoising process. Instead of the traditional uniform distribution-based time step sampling, we introduce a Beta distribution-like sampling technique that prioritizes critical steps in the early and late stages of the process. Our hypothesis is that certain steps exhibit significant changes in image content, while others contribute minimally. We validated our approach using Fourier transforms to measure frequency response changes at each step, revealing substantial low-frequency changes early on and high-frequency adjustments later. Experiments with ADM and Stable Diffusion demonstrated that our Beta Sampling method consistently outperforms uniform sampling, achieving better FID and IS scores, and offers competitive efficiency relative to state-of-the-art methods like AutoDiffusion. This work provides a practical framework for enhancing diffusion model efficiency by focusing computational resources on the most impactful steps, with potential for further optimization and broader application.

Denoising Diffusion Models (DDMs) have become a popular tool for generating high-quality samples from complex data distributions. These models are able to capture sophisticated patterns and structures in the data, and can generate samples that are highly diverse and representative of the underlying distribution. However, one of the main limitations of diffusion models is the complexity of sample generation, since a large number of inference timesteps is required to faithfully capture the data distribution. In this paper, we present MMD-DDM, a novel method for fast sampling of diffusion models. Our approach is based on the idea of using the Maximum Mean Discrepancy (MMD) to finetune the learned distribution with a given budget of timesteps. This allows the finetuned model to significantly improve the speed-quality trade-off, by substantially increasing fidelity in inference regimes with few steps or, equivalently, by reducing the required number of steps to reach a target fidelity, thus paving the way for a more practical adoption of diffusion models in a wide range of applications. We evaluate our approach on unconditional image generation with extensive experiments across the CIFAR-10, CelebA, ImageNet and LSUN-Church datasets. Our findings show that the proposed method is able to produce high-quality samples in a fraction of the time required by widely-used diffusion models, and outperforms state-of-the-art techniques for accelerated sampling. Code is available at: https://github.com/diegovalsesia/MMD-DDM.

Diffusion models have found widespread adoption in various areas. However, their sampling process is slow because it requires hundreds to thousands of network evaluations to emulate a continuous process defined by differential equations. In this work, we use neural operators, an efficient method to solve the probability flow differential equations, to accelerate the sampling process of diffusion models. Compared to other fast sampling methods that have a sequential nature, we are the first to propose parallel decoding method that generates images with only one model forward pass. We propose diffusion model sampling with neural operator (DSNO) that maps the initial condition, i.e., Gaussian distribution, to the continuous-time solution trajectory of the reverse diffusion process. To model the temporal correlations along the trajectory, we introduce temporal convolution layers that are parameterized in the Fourier space into the given diffusion model backbone. We show our method achieves state-of-the-art FID of 4.12 for CIFAR-10 and 8.35 for ImageNet-64 in the one-model-evaluation setting.

Diffusion models excel at joint pixel sampling for image generation but lack efficient training-free methods for partial conditional sampling (e.g., inpainting with known pixels). Prior work typically formulates this as an intractable inverse problem, relying on coarse variational approximations, heuristic losses requiring expensive backpropagation, or slow stochastic sampling. These limitations preclude: (1) accurate distributional matching in inpainting results, (2) efficient inference modes without gradient, (3) compatibility with fast ODE-based samplers. To address these limitations, we propose LanPaint: a training-free, asymptotically exact partial conditional sampling methods for ODE-based and rectified flow diffusion models. By leveraging carefully designed Langevin dynamics, LanPaint enables fast, backpropagation-free Monte Carlo sampling. Experiments demonstrate that our approach achieves superior performance with precise partial conditioning and visually coherent inpainting across diverse tasks.

Diffusion models have recently emerged as powerful generative priors for solving inverse problems. However, training diffusion models in the pixel space are both data-intensive and computationally demanding, which restricts their applicability as priors for high-dimensional real-world data such as medical images. Latent diffusion models, which operate in a much lower-dimensional space, offer a solution to these challenges. However, incorporating latent diffusion models to solve inverse problems remains a challenging problem due to the nonlinearity of the encoder and decoder. To address these issues, we propose ReSample, an algorithm that can solve general inverse problems with pre-trained latent diffusion models. Our algorithm incorporates data consistency by solving an optimization problem during the reverse sampling process, a concept that we term as hard data consistency. Upon solving this optimization problem, we propose a novel resampling scheme to map the measurement-consistent sample back onto the noisy data manifold and theoretically demonstrate its benefits. Lastly, we apply our algorithm to solve a wide range of linear and nonlinear inverse problems in both natural and medical images, demonstrating that our approach outperforms existing state-of-the-art approaches, including those based on pixel-space diffusion models.

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce ``Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Diffusion models show promising generation capability for a variety of data. Despite their high generation quality, the inference for diffusion models is still time-consuming due to the numerous sampling iterations required. To accelerate the inference, we propose ReDi, a simple yet learning-free Retrieval-based Diffusion sampling framework. From a precomputed knowledge base, ReDi retrieves a trajectory similar to the partially generated trajectory at an early stage of generation, skips a large portion of intermediate steps, and continues sampling from a later step in the retrieved trajectory. We theoretically prove that the generation performance of ReDi is guaranteed. Our experiments demonstrate that ReDi improves the model inference efficiency by 2x speedup. Furthermore, ReDi is able to generalize well in zero-shot cross-domain image generation such as image stylization.

In applications of diffusion models, controllable generation is of practical significance, but is also challenging. Current methods for controllable generation primarily focus on modifying the score function of diffusion models, while Mean Reverting (MR) Diffusion directly modifies the structure of the stochastic differential equation (SDE), making the incorporation of image conditions simpler and more natural. However, current training-free fast samplers are not directly applicable to MR Diffusion. And thus MR Diffusion requires hundreds of NFEs (number of function evaluations) to obtain high-quality samples. In this paper, we propose a new algorithm named MRS (MR Sampler) to reduce the sampling NFEs of MR Diffusion. We solve the reverse-time SDE and the probability flow ordinary differential equation (PF-ODE) associated with MR Diffusion, and derive semi-analytical solutions. The solutions consist of an analytical function and an integral parameterized by a neural network. Based on this solution, we can generate high-quality samples in fewer steps. Our approach does not require training and supports all mainstream parameterizations, including noise prediction, data prediction and velocity prediction. Extensive experiments demonstrate that MR Sampler maintains high sampling quality with a speedup of 10 to 20 times across ten different image restoration tasks. Our algorithm accelerates the sampling procedure of MR Diffusion, making it more practical in controllable generation.

The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires hundreds to thousands of time discretization steps of the learned diffusion process to reach the desired accuracy. Our goal is to develop a fast sampling method for DMs with a much less number of steps while retaining high sample quality. To this end, we systematically analyze the sampling procedure in DMs and identify key factors that affect the sample quality, among which the method of discretization is most crucial. By carefully examining the learned diffusion process, we propose Diffusion Exponential Integrator Sampler~(DEIS). It is based on the Exponential Integrator designed for discretizing ordinary differential equations (ODEs) and leverages a semilinear structure of the learned diffusion process to reduce the discretization error. The proposed method can be applied to any DMs and can generate high-fidelity samples in as few as 10 steps. In our experiments, it takes about 3 minutes on one A6000 GPU to generate 50k images from CIFAR10. Moreover, by directly using pre-trained DMs, we achieve the state-of-art sampling performance when the number of score function evaluation~(NFE) is limited, e.g., 4.17 FID with 10 NFEs, 3.37 FID, and 9.74 IS with only 15 NFEs on CIFAR10. Code is available at https://github.com/qsh-zh/deis

Score-based diffusion models learn to reverse a stochastic differential equation that maps data to noise. However, for complex tasks, numerical error can compound and result in highly unnatural samples. Previous work mitigates this drift with thresholding, which projects to the natural data domain (such as pixel space for images) after each diffusion step, but this leads to a mismatch between the training and generative processes. To incorporate data constraints in a principled manner, we present Reflected Diffusion Models, which instead reverse a reflected stochastic differential equation evolving on the support of the data. Our approach learns the perturbed score function through a generalized score matching loss and extends key components of standard diffusion models including diffusion guidance, likelihood-based training, and ODE sampling. We also bridge the theoretical gap with thresholding: such schemes are just discretizations of reflected SDEs. On standard image benchmarks, our method is competitive with or surpasses the state of the art without architectural modifications and, for classifier-free guidance, our approach enables fast exact sampling with ODEs and produces more faithful samples under high guidance weight.

Despite the remarkable success of diffusion models in image generation, slow sampling remains a persistent issue. To accelerate the sampling process, prior studies have reformulated diffusion sampling as an ODE/SDE and introduced higher-order numerical methods. However, these methods often produce divergence artifacts, especially with a low number of sampling steps, which limits the achievable acceleration. In this paper, we investigate the potential causes of these artifacts and suggest that the small stability regions of these methods could be the principal cause. To address this issue, we propose two novel techniques. The first technique involves the incorporation of Heavy Ball (HB) momentum, a well-known technique for improving optimization, into existing diffusion numerical methods to expand their stability regions. We also prove that the resulting methods have first-order convergence. The second technique, called Generalized Heavy Ball (GHVB), constructs a new high-order method that offers a variable trade-off between accuracy and artifact suppression. Experimental results show that our techniques are highly effective in reducing artifacts and improving image quality, surpassing state-of-the-art diffusion solvers on both pixel-based and latent-based diffusion models for low-step sampling. Our research provides novel insights into the design of numerical methods for future diffusion work.

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.

Diffusion-based image super-resolution (SR) methods have achieved remarkable success by leveraging large pre-trained text-to-image diffusion models as priors. However, these methods still face two challenges: the requirement for dozens of sampling steps to achieve satisfactory results, which limits efficiency in real scenarios, and the neglect of degradation models, which are critical auxiliary information in solving the SR problem. In this work, we introduced a novel one-step SR model, which significantly addresses the efficiency issue of diffusion-based SR methods. Unlike existing fine-tuning strategies, we designed a degradation-guided Low-Rank Adaptation (LoRA) module specifically for SR, which corrects the model parameters based on the pre-estimated degradation information from low-resolution images. This module not only facilitates a powerful data-dependent or degradation-dependent SR model but also preserves the generative prior of the pre-trained diffusion model as much as possible. Furthermore, we tailor a novel training pipeline by introducing an online negative sample generation strategy. Combined with the classifier-free guidance strategy during inference, it largely improves the perceptual quality of the super-resolution results. Extensive experiments have demonstrated the superior efficiency and effectiveness of the proposed model compared to recent state-of-the-art methods.

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Denoising diffusion probabilistic models (DDPMs) have achieved high quality image generation without adversarial training, yet they require simulating a Markov chain for many steps to produce a sample. To accelerate sampling, we present denoising diffusion implicit models (DDIMs), a more efficient class of iterative implicit probabilistic models with the same training procedure as DDPMs. In DDPMs, the generative process is defined as the reverse of a Markovian diffusion process. We construct a class of non-Markovian diffusion processes that lead to the same training objective, but whose reverse process can be much faster to sample from. We empirically demonstrate that DDIMs can produce high quality samples 10 times to 50 times faster in terms of wall-clock time compared to DDPMs, allow us to trade off computation for sample quality, and can perform semantically meaningful image interpolation directly in the latent space.

Generative diffusion models provide strong priors for text-to-image generation and thereby serve as a foundation for conditional generation tasks such as image editing, restoration, and super-resolution. However, one major limitation of diffusion models is their slow sampling time. To address this challenge, we present a novel conditional distillation method designed to supplement the diffusion priors with the help of image conditions, allowing for conditional sampling with very few steps. We directly distill the unconditional pre-training in a single stage through joint-learning, largely simplifying the previous two-stage procedures that involve both distillation and conditional finetuning separately. Furthermore, our method enables a new parameter-efficient distillation mechanism that distills each task with only a small number of additional parameters combined with the shared frozen unconditional backbone. Experiments across multiple tasks including super-resolution, image editing, and depth-to-image generation demonstrate that our method outperforms existing distillation techniques for the same sampling time. Notably, our method is the first distillation strategy that can match the performance of the much slower fine-tuned conditional diffusion models.

We propose a Monte Carlo sampler from the reverse diffusion process. Unlike the practice of diffusion models, where the intermediary updates -- the score functions -- are learned with a neural network, we transform the score matching problem into a mean estimation one. By estimating the means of the regularized posterior distributions, we derive a novel Monte Carlo sampling algorithm called reverse diffusion Monte Carlo (rdMC), which is distinct from the Markov chain Monte Carlo (MCMC) methods. We determine the sample size from the error tolerance and the properties of the posterior distribution to yield an algorithm that can approximately sample the target distribution with any desired accuracy. Additionally, we demonstrate and prove under suitable conditions that sampling with rdMC can be significantly faster than that with MCMC. For multi-modal target distributions such as those in Gaussian mixture models, rdMC greatly improves over the Langevin-style MCMC sampling methods both theoretically and in practice. The proposed rdMC method offers a new perspective and solution beyond classical MCMC algorithms for the challenging complex distributions.

Denoising Diffusion Probabilistic Models have shown an impressive generation quality, although their long sampling chain leads to high computational costs. In this paper, we observe that a long sampling chain also leads to an error accumulation phenomenon, which is similar to the exposure bias problem in autoregressive text generation. Specifically, we note that there is a discrepancy between training and testing, since the former is conditioned on the ground truth samples, while the latter is conditioned on the previously generated results. To alleviate this problem, we propose a very simple but effective training regularization, consisting in perturbing the ground truth samples to simulate the inference time prediction errors. We empirically show that, without affecting the recall and precision, the proposed input perturbation leads to a significant improvement in the sample quality while reducing both the training and the inference times. For instance, on CelebA 64times64, we achieve a new state-of-the-art FID score of 1.27, while saving 37.5% of the training time. The code is publicly available at https://github.com/forever208/DDPM-IP

Diffusion models have emerged as powerful tools for generative tasks, producing high-quality outputs across diverse domains. However, how the generated data responds to the initial noise perturbation in diffusion models remains under-explored, which hinders understanding the controllability of the sampling process. In this work, we first observe an interesting phenomenon: the relationship between the change of generation outputs and the scale of initial noise perturbation is highly linear through the diffusion ODE sampling. Then we provide both theoretical and empirical study to justify this linearity property of this input-output (noise-generation data) relationship. Inspired by these new insights, we propose a novel Controllable and Constrained Sampling method (CCS) together with a new controller algorithm for diffusion models to sample with desired statistical properties while preserving good sample quality. We perform extensive experiments to compare our proposed sampling approach with other methods on both sampling controllability and sampled data quality. Results show that our CCS method achieves more precisely controlled sampling while maintaining superior sample quality and diversity.

This monograph presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward process that gradually corrupts data into noise, linking the data distribution to a simple prior through a continuum of intermediate distributions. The goal is to learn a reverse process that transforms noise back into data while recovering the same intermediates. We describe three complementary views. The variational view, inspired by variational autoencoders, sees diffusion as learning to remove noise step by step. The score-based view, rooted in energy-based modeling, learns the gradient of the evolving data distribution, indicating how to nudge samples toward more likely regions. The flow-based view, related to normalizing flows, treats generation as following a smooth path that moves samples from noise to data under a learned velocity field. These perspectives share a common backbone: a time-dependent velocity field whose flow transports a simple prior to the data. Sampling then amounts to solving a differential equation that evolves noise into data along a continuous trajectory. On this foundation, the monograph discusses guidance for controllable generation, efficient numerical solvers, and diffusion-motivated flow-map models that learn direct mappings between arbitrary times. It provides a conceptual and mathematically grounded understanding of diffusion models for readers with basic deep-learning knowledge.

Diffusion-based generative models use stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models. We characterize an implicit denoising trajectory and discuss its vital role in forming the coupled sampling trajectory with a strong shape regularity, regardless of the generated content. We also describe a dynamic programming-based scheme to make the time schedule in sampling better fit the underlying trajectory structure. This simple strategy requires minimal modification to any given ODE-based numerical solvers and incurs negligible computational cost, while delivering superior performance in image generation, especially in 5sim 10 function evaluations.

Recent advancements in text-guided diffusion models have unlocked powerful image manipulation capabilities. However, applying these methods to real images necessitates the inversion of the images into the domain of the pretrained diffusion model. Achieving faithful inversion remains a challenge, particularly for more recent models trained to generate images with a small number of denoising steps. In this work, we introduce an inversion method with a high quality-to-operation ratio, enhancing reconstruction accuracy without increasing the number of operations. Building on reversing the diffusion sampling process, our method employs an iterative renoising mechanism at each inversion sampling step. This mechanism refines the approximation of a predicted point along the forward diffusion trajectory, by iteratively applying the pretrained diffusion model, and averaging these predictions. We evaluate the performance of our ReNoise technique using various sampling algorithms and models, including recent accelerated diffusion models. Through comprehensive evaluations and comparisons, we show its effectiveness in terms of both accuracy and speed. Furthermore, we confirm that our method preserves editability by demonstrating text-driven image editing on real images.

While diffusion models have made remarkable progress in image generation, their outputs can still appear unrealistic and lack fine details, especially when using fewer number of neural function evaluations (NFEs) or lower guidance scales. To address this issue, we propose a novel momentum-based sampling technique, termed history-guided sampling (HiGS), which enhances quality and efficiency of diffusion sampling by integrating recent model predictions into each inference step. Specifically, HiGS leverages the difference between the current prediction and a weighted average of past predictions to steer the sampling process toward more realistic outputs with better details and structure. Our approach introduces practically no additional computation and integrates seamlessly into existing diffusion frameworks, requiring neither extra training nor fine-tuning. Extensive experiments show that HiGS consistently improves image quality across diverse models and architectures and under varying sampling budgets and guidance scales. Moreover, using a pretrained SiT model, HiGS achieves a new state-of-the-art FID of 1.61 for unguided ImageNet generation at 256times256 with only 30 sampling steps (instead of the standard 250). We thus present HiGS as a plug-and-play enhancement to standard diffusion sampling that enables faster generation with higher fidelity.

Diffusion models may be formulated as a time-indexed sequence of energy-based models, where the score corresponds to the negative gradient of an energy function. As opposed to learning the score directly, an energy parameterization is attractive as the energy itself can be used to control generation via Monte Carlo samplers. Architectural constraints and training instability in energy parameterized models have so far yielded inferior performance compared to directly approximating the score or denoiser. We address these deficiencies by introducing a novel training regime for the energy function through distillation of pre-trained diffusion models, resembling a Helmholtz decomposition of the score vector field. We further showcase the synergies between energy and score by casting the diffusion sampling procedure as a Feynman Kac model where sampling is controlled using potentials from the learnt energy functions. The Feynman Kac model formalism enables composition and low temperature sampling through sequential Monte Carlo.

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS

Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active guidance towards task-desired properties. Despite the significant empirical success, theory of diffusion models is very limited, potentially slowing down principled methodological innovations for further harnessing and improving diffusion models. In this paper, we review emerging applications of diffusion models, understanding their sample generation under various controls. Next, we overview the existing theories of diffusion models, covering their statistical properties and sampling capabilities. We adopt a progressive routine, beginning with unconditional diffusion models and connecting to conditional counterparts. Further, we review a new avenue in high-dimensional structured optimization through conditional diffusion models, where searching for solutions is reformulated as a conditional sampling problem and solved by diffusion models. Lastly, we discuss future directions about diffusion models. The purpose of this paper is to provide a well-rounded theoretical exposure for stimulating forward-looking theories and methods of diffusion models.

Diffusion models have demonstrated impressive generative capabilities, but their exposure bias problem, described as the input mismatch between training and sampling, lacks in-depth exploration. In this paper, we systematically investigate the exposure bias problem in diffusion models by first analytically modelling the sampling distribution, based on which we then attribute the prediction error at each sampling step as the root cause of the exposure bias issue. Furthermore, we discuss potential solutions to this issue and propose an intuitive metric for it. Along with the elucidation of exposure bias, we propose a simple, yet effective, training-free method called Epsilon Scaling to alleviate the exposure bias. We show that Epsilon Scaling explicitly moves the sampling trajectory closer to the vector field learned in the training phase by scaling down the network output (Epsilon), mitigating the input mismatch between training and sampling. Experiments on various diffusion frameworks (ADM, DDPM/DDIM, EDM, LDM), unconditional and conditional settings, and deterministic vs. stochastic sampling verify the effectiveness of our method. Remarkably, our ADM-ES, as a SOTA stochastic sampler, obtains 2.17 FID on CIFAR-10 under 100-step unconditional generation. The code is available at https://github.com/forever208/ADM-ES and https://github.com/forever208/EDM-ES.

Score-based diffusion models are a class of generative models whose dynamics is described by stochastic differential equations that map noise into data. While recent works have started to lay down a theoretical foundation for these models, an analytical understanding of the role of the diffusion time T is still lacking. Current best practice advocates for a large T to ensure that the forward dynamics brings the diffusion sufficiently close to a known and simple noise distribution; however, a smaller value of T should be preferred for a better approximation of the score-matching objective and higher computational efficiency. Starting from a variational interpretation of diffusion models, in this work we quantify this trade-off, and suggest a new method to improve quality and efficiency of both training and sampling, by adopting smaller diffusion times. Indeed, we show how an auxiliary model can be used to bridge the gap between the ideal and the simulated forward dynamics, followed by a standard reverse diffusion process. Empirical results support our analysis; for image data, our method is competitive w.r.t. the state-of-the-art, according to standard sample quality metrics and log-likelihood.

Diffusion Probabilistic Models (DPM) have shown remarkable efficacy in the synthesis of high-quality images. However, their inference process characteristically requires numerous, potentially hundreds, of iterative steps, which could exaggerate the problem of exposure bias due to the training and inference discrepancy. Previous work has attempted to mitigate this issue by perturbing inputs during training, which consequently mandates the retraining of the DPM. In this work, we conduct a systematic study of exposure bias in DPM and, intriguingly, we find that the exposure bias could be alleviated with a novel sampling method that we propose, without retraining the model. We empirically and theoretically show that, during inference, for each backward time step t and corresponding state x_t, there might exist another time step t_s which exhibits superior coupling with x_t. Based on this finding, we introduce a sampling method named Time-Shift Sampler. Our framework can be seamlessly integrated to existing sampling algorithms, such as DDPM, DDIM and other high-order solvers, inducing merely minimal additional computations. Experimental results show our method brings significant and consistent improvements in FID scores on different datasets and sampling methods. For example, integrating Time-Shift Sampler to F-PNDM yields a FID=3.88, achieving 44.49\% improvements as compared to F-PNDM, on CIFAR-10 with 10 sampling steps, which is more performant than the vanilla DDIM with 100 sampling steps. Our code is available at https://github.com/Mingxiao-Li/TS-DPM.

We provide a framework for solving inverse problems with diffusion models learned from linearly corrupted data. Our method, Ambient Diffusion Posterior Sampling (A-DPS), leverages a generative model pre-trained on one type of corruption (e.g. image inpainting) to perform posterior sampling conditioned on measurements from a potentially different forward process (e.g. image blurring). We test the efficacy of our approach on standard natural image datasets (CelebA, FFHQ, and AFHQ) and we show that A-DPS can sometimes outperform models trained on clean data for several image restoration tasks in both speed and performance. We further extend the Ambient Diffusion framework to train MRI models with access only to Fourier subsampled multi-coil MRI measurements at various acceleration factors (R=2, 4, 6, 8). We again observe that models trained on highly subsampled data are better priors for solving inverse problems in the high acceleration regime than models trained on fully sampled data. We open-source our code and the trained Ambient Diffusion MRI models: https://github.com/utcsilab/ambient-diffusion-mri .

Denoising diffusion models have emerged as the go-to framework for solving inverse problems in imaging. A critical concern regarding these models is their performance on out-of-distribution (OOD) tasks, which remains an under-explored challenge. Realistic reconstructions inconsistent with the measured data can be generated, hallucinating image features that are uniquely present in the training dataset. To simultaneously enforce data-consistency and leverage data-driven priors, we introduce a novel sampling framework called Steerable Conditional Diffusion. This framework adapts the denoising network specifically to the available measured data. Utilising our proposed method, we achieve substantial enhancements in OOD performance across diverse imaging modalities, advancing the robust deployment of denoising diffusion models in real-world applications.

Diffusion models trained with mean squared error loss tend to generate unrealistic samples. Current state-of-the-art models rely on classifier-free guidance to improve sample quality, yet its surprising effectiveness is not fully understood. In this paper, We show that the effectiveness of classifier-free guidance partly originates from it being a form of implicit perceptual guidance. As a result, we can directly incorporate perceptual loss in diffusion training to improve sample quality. Since the score matching objective used in diffusion training strongly resembles the denoising autoencoder objective used in unsupervised training of perceptual networks, the diffusion model itself is a perceptual network and can be used to generate meaningful perceptual loss. We propose a novel self-perceptual objective that results in diffusion models capable of generating more realistic samples. For conditional generation, our method only improves sample quality without entanglement with the conditional input and therefore does not sacrifice sample diversity. Our method can also improve sample quality for unconditional generation, which was not possible with classifier-free guidance before.

Diffusion models have emerged as a pivotal advancement in generative models, setting new standards to the quality of the generated instances. In the current paper we aim to underscore a discrepancy between conventional training methods and the desired conditional sampling behavior of these models. While the prevalent classifier-free guidance technique works well, it's not without flaws. At higher values for the guidance scale parameter w, we often get out of distribution samples and mode collapse, whereas at lower values for w we may not get the desired specificity. To address these challenges, we introduce an updated loss function that better aligns training objectives with sampling behaviors. Experimental validation with FID scores on CIFAR-10 elucidates our method's ability to produce higher quality samples with fewer sampling timesteps, and be more robust to the choice of guidance scale w. We also experiment with fine-tuning Stable Diffusion on the proposed loss, to provide early evidence that large diffusion models may also benefit from this refined loss function.

Recent years have witnessed significant progress in developing efficient training and fast sampling approaches for diffusion models. A recent remarkable advancement is the use of stochastic differential equations (SDEs) to describe data perturbation and generative modeling in a unified mathematical framework. In this paper, we reveal several intriguing geometric structures of diffusion models and contribute a simple yet powerful interpretation to their sampling dynamics. Through carefully inspecting a popular variance-exploding SDE and its marginal-preserving ordinary differential equation (ODE) for sampling, we discover that the data distribution and the noise distribution are smoothly connected with an explicit, quasi-linear sampling trajectory, and another implicit denoising trajectory, which even converges faster in terms of visual quality. We also establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm, with which we can characterize the asymptotic behavior of diffusion models and identify the score deviation. These new geometric observations enable us to improve previous sampling algorithms, re-examine latent interpolation, as well as re-explain the working principles of distillation-based fast sampling techniques.

Diffusion models (DMs) have established themselves as the state-of-the-art generative modeling approach in the visual domain and beyond. A crucial drawback of DMs is their slow sampling speed, relying on many sequential function evaluations through large neural networks. Sampling from DMs can be seen as solving a differential equation through a discretized set of noise levels known as the sampling schedule. While past works primarily focused on deriving efficient solvers, little attention has been given to finding optimal sampling schedules, and the entire literature relies on hand-crafted heuristics. In this work, for the first time, we propose a general and principled approach to optimizing the sampling schedules of DMs for high-quality outputs, called Align Your Steps. We leverage methods from stochastic calculus and find optimal schedules specific to different solvers, trained DMs and datasets. We evaluate our novel approach on several image, video as well as 2D toy data synthesis benchmarks, using a variety of different samplers, and observe that our optimized schedules outperform previous hand-crafted schedules in almost all experiments. Our method demonstrates the untapped potential of sampling schedule optimization, especially in the few-step synthesis regime.

Discrete diffusion models are often trained through clean-data prediction, but the prediction can be used in different ways to define the reverse dynamics. In Masked Diffusion Models (MDM) these choices largely coincide, whereas in Uniform Diffusion Models (UDM) they do not. We show that the standard plug-in bridge parameterization for UDM is not optimized by the denoising posterior, but by a leave-one-out posterior that predicts each clean token without using its own noisy observation. This identifies a mismatch between the plug-in ELBO and the usual cross-entropy denoising objective. We characterize the leave-one-out target and derive exact conversions between the denoiser, the leave-one-out posterior, and the score. These conversions allow us to disentangle parameterization and training objective. Our results also lead to inference improvements without any additional training through an informed predictor-corrector sampler and improved temperature sampling based on the leave-one-out predictor. We further introduce an absorbing-state reformulation of uniform diffusion that preserves the UDM joint law while decomposing it into masked-diffusion-like sampling operations, with simpler denoising posteriors, carry-over unmasking, and a natural remasking mechanism. On language modeling, leave-one-out parameterizations consistently improve UDM generation, while the absorbing construction matches or surpasses masked diffusion. These results suggest that the empirical gap between masked and uniform diffusion is driven less by the choice of marginals themselves than by parameterization and sampling design. The code and models can be found at https://github.com/samsongourevitch/rev_udm.

Score-based diffusion models generate samples from an unknown target distribution using a time-reversed diffusion process. While such models represent state-of-the-art approaches in industrial applications such as artificial image generation, it has recently been noted that their performance can be further improved by considering injection noise with heavy tailed characteristics. Here, I present a generalization of generative diffusion processes to a wide class of non-Gaussian noise processes. I consider forward processes driven by standard Gaussian noise with super-imposed Poisson jumps representing a finite activity Levy process. The generative process is shown to be governed by a generalized score function that depends on the jump amplitude distribution and can be estimated by minimizing a simple MSE loss as in conventional Gaussian models. Both probability flow ODE and SDE formulations are derived using basic technical effort. A detailed implementation for a pure jump process with Laplace distributed amplitudes yields a generalized score function in closed analytical form and is shown to outperform the equivalent Gaussian model in specific parameter regimes.

Diffusion probabilistic models (DPMs) have achieved impressive success in high-resolution image synthesis, especially in recent large-scale text-to-image generation applications. An essential technique for improving the sample quality of DPMs is guided sampling, which usually needs a large guidance scale to obtain the best sample quality. The commonly-used fast sampler for guided sampling is DDIM, a first-order diffusion ODE solver that generally needs 100 to 250 steps for high-quality samples. Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance, their effectiveness for guided sampling has not been well-tested before. In this work, we demonstrate that previous high-order fast samplers suffer from instability issues, and they even become slower than DDIM when the guidance scale grows large. To further speed up guided sampling, we propose DPM-Solver++, a high-order solver for the guided sampling of DPMs. DPM-Solver++ solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution. We further propose a multistep variant of DPM-Solver++ to address the instability issue by reducing the effective step size. Experiments show that DPM-Solver++ can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.

Classifier-free guided diffusion models have recently been shown to be highly effective at high-resolution image generation, and they have been widely used in large-scale diffusion frameworks including DALLE-2, Stable Diffusion and Imagen. However, a downside of classifier-free guided diffusion models is that they are computationally expensive at inference time since they require evaluating two diffusion models, a class-conditional model and an unconditional model, tens to hundreds of times. To deal with this limitation, we propose an approach to distilling classifier-free guided diffusion models into models that are fast to sample from: Given a pre-trained classifier-free guided model, we first learn a single model to match the output of the combined conditional and unconditional models, and then we progressively distill that model to a diffusion model that requires much fewer sampling steps. For standard diffusion models trained on the pixel-space, our approach is able to generate images visually comparable to that of the original model using as few as 4 sampling steps on ImageNet 64x64 and CIFAR-10, achieving FID/IS scores comparable to that of the original model while being up to 256 times faster to sample from. For diffusion models trained on the latent-space (e.g., Stable Diffusion), our approach is able to generate high-fidelity images using as few as 1 to 4 denoising steps, accelerating inference by at least 10-fold compared to existing methods on ImageNet 256x256 and LAION datasets. We further demonstrate the effectiveness of our approach on text-guided image editing and inpainting, where our distilled model is able to generate high-quality results using as few as 2-4 denoising steps.

Online reinforcement learning (RL) has been central to post-training language models, but its extension to diffusion models remains challenging due to intractable likelihoods. Recent works discretize the reverse sampling process to enable GRPO-style training, yet they inherit fundamental drawbacks, including solver restrictions, forward-reverse inconsistency, and complicated integration with classifier-free guidance (CFG). We introduce Diffusion Negative-aware FineTuning (DiffusionNFT), a new online RL paradigm that optimizes diffusion models directly on the forward process via flow matching. DiffusionNFT contrasts positive and negative generations to define an implicit policy improvement direction, naturally incorporating reinforcement signals into the supervised learning objective. This formulation enables training with arbitrary black-box solvers, eliminates the need for likelihood estimation, and requires only clean images rather than sampling trajectories for policy optimization. DiffusionNFT is up to 25times more efficient than FlowGRPO in head-to-head comparisons, while being CFG-free. For instance, DiffusionNFT improves the GenEval score from 0.24 to 0.98 within 1k steps, while FlowGRPO achieves 0.95 with over 5k steps and additional CFG employment. By leveraging multiple reward models, DiffusionNFT significantly boosts the performance of SD3.5-Medium in every benchmark tested.

Diffusion models have gained prominence in generating high-quality sequences of text. Nevertheless, current approaches predominantly represent discrete text within a continuous diffusion space, which incurs substantial computational overhead during training and results in slower sampling speeds. In this paper, we introduce a soft absorbing state that facilitates the diffusion model in learning to reconstruct discrete mutations based on the underlying Gaussian space, thereby enhancing its capacity to recover conditional signals. During the sampling phase, we employ state-of-the-art ODE solvers within the continuous space to expedite the sampling process. Comprehensive experimental evaluations reveal that our proposed method effectively accelerates the training convergence by 4x and generates samples of similar quality 800x faster, rendering it significantly closer to practical application. The code is released at \url{https://github.com/Shark-NLP/DiffuSeq

Diffusion probabilistic models (DPMs) have achieved impressive success in visual generation. While, they suffer from slow inference speed due to iterative sampling. Employing fewer sampling steps is an intuitive solution, but this will also introduces discretization error. Existing fast samplers make inspiring efforts to reduce discretization error through the adoption of high-order solvers, potentially reaching a plateau in terms of optimization. This raises the question: can the sampling process be accelerated further? In this paper, we re-examine the nature of sampling errors, discerning that they comprise two distinct elements: the widely recognized discretization error and the less explored approximation error. Our research elucidates the dynamics between these errors and the step by implementing a dual-error disentanglement strategy. Building on these foundations, we introduce an unified and training-free acceleration framework, DualFast, designed to enhance the speed of DPM sampling by concurrently accounting for both error types, thereby minimizing the total sampling error. DualFast is seamlessly compatible with existing samplers and significantly boost their sampling quality and speed, particularly in extremely few sampling steps. We substantiate the effectiveness of our framework through comprehensive experiments, spanning both unconditional and conditional sampling domains, across both pixel-space and latent-space DPMs.

Diffusion models have opened the path to a wide range of text-based image editing frameworks. However, these typically build on the multi-step nature of the diffusion backwards process, and adapting them to distilled, fast-sampling methods has proven surprisingly challenging. Here, we focus on a popular line of text-based editing frameworks - the ``edit-friendly'' DDPM-noise inversion approach. We analyze its application to fast sampling methods and categorize its failures into two classes: the appearance of visual artifacts, and insufficient editing strength. We trace the artifacts to mismatched noise statistics between inverted noises and the expected noise schedule, and suggest a shifted noise schedule which corrects for this offset. To increase editing strength, we propose a pseudo-guidance approach that efficiently increases the magnitude of edits without introducing new artifacts. All in all, our method enables text-based image editing with as few as three diffusion steps, while providing novel insights into the mechanisms behind popular text-based editing approaches.

We discover that common diffusion noise schedules do not enforce the last timestep to have zero signal-to-noise ratio (SNR), and some implementations of diffusion samplers do not start from the last timestep. Such designs are flawed and do not reflect the fact that the model is given pure Gaussian noise at inference, creating a discrepancy between training and inference. We show that the flawed design causes real problems in existing implementations. In Stable Diffusion, it severely limits the model to only generate images with medium brightness and prevents it from generating very bright and dark samples. We propose a few simple fixes: (1) rescale the noise schedule to enforce zero terminal SNR; (2) train the model with v prediction; (3) change the sampler to always start from the last timestep; (4) rescale classifier-free guidance to prevent over-exposure. These simple changes ensure the diffusion process is congruent between training and inference and allow the model to generate samples more faithful to the original data distribution.

Coarse-guided visual generation, which synthesizes fine visual samples from degraded or low-fidelity coarse references, is essential for various real-world applications. While training-based approaches are effective, they are inherently limited by high training costs and restricted generalization due to paired data collection. Accordingly, recent training-free works propose to leverage pretrained diffusion models and incorporate guidance during the sampling process. However, these training-free methods either require knowing the forward (fine-to-coarse) transformation operator, e.g., bicubic downsampling, or are difficult to balance between guidance and synthetic quality. To address these challenges, we propose a novel guided method by using the h-transform, a tool that can constrain stochastic processes (e.g., sampling process) under desired conditions. Specifically, we modify the transition probability at each sampling timestep by adding to the original differential equation with a drift function, which approximately steers the generation toward the ideal fine sample. To address unavoidable approximation errors, we introduce a noise-level-aware schedule that gradually de-weights the term as the error increases, ensuring both guidance adherence and high-quality synthesis. Extensive experiments across diverse image and video generation tasks demonstrate the effectiveness and generalization of our method.

A common recipe to improve diffusion models at test-time so that samples score highly against a user-specified reward is to introduce the gradient of the reward into the dynamics of the diffusion itself. This procedure is often ill posed, as user-specified rewards are usually only well defined on the data distribution at the end of generation. While common workarounds to this problem are to use a denoiser to estimate what a sample would have been at the end of generation, we propose a simple solution to this problem by working directly with a flow map. By exploiting a relationship between the flow map and velocity field governing the instantaneous transport, we construct an algorithm, Flow Map Trajectory Tilting (FMTT), which provably performs better ascent on the reward than standard test-time methods involving the gradient of the reward. The approach can be used to either perform exact sampling via importance weighting or principled search that identifies local maximizers of the reward-tilted distribution. We demonstrate the efficacy of our approach against other look-ahead techniques, and show how the flow map enables engagement with complicated reward functions that make possible new forms of image editing, e.g. by interfacing with vision language models.

We study the problem of posterior sampling in discrete-state spaces using discrete diffusion models. While posterior sampling methods for continuous diffusion models have achieved remarkable progress, analogous methods for discrete diffusion models remain challenging. In this work, we introduce a principled plug-and-play discrete diffusion posterior sampling algorithm based on split Gibbs sampling, which we call SG-DPS. Our algorithm enables reward-guided generation and solving inverse problems in discrete-state spaces. We demonstrate that SG-DPS converges to the true posterior distribution on synthetic benchmarks, and enjoys state-of-the-art posterior sampling performance on a range of benchmarks for discrete data, achieving up to 2x improved performance compared to existing baselines.

Colloquially speaking, image generation models based upon diffusion processes are frequently said to exhibit "hallucinations," samples that could never occur in the training data. But where do such hallucinations come from? In this paper, we study a particular failure mode in diffusion models, which we term mode interpolation. Specifically, we find that diffusion models smoothly "interpolate" between nearby data modes in the training set, to generate samples that are completely outside the support of the original training distribution; this phenomenon leads diffusion models to generate artifacts that never existed in real data (i.e., hallucinations). We systematically study the reasons for, and the manifestation of this phenomenon. Through experiments on 1D and 2D Gaussians, we show how a discontinuous loss landscape in the diffusion model's decoder leads to a region where any smooth approximation will cause such hallucinations. Through experiments on artificial datasets with various shapes, we show how hallucination leads to the generation of combinations of shapes that never existed. Finally, we show that diffusion models in fact know when they go out of support and hallucinate. This is captured by the high variance in the trajectory of the generated sample towards the final few backward sampling process. Using a simple metric to capture this variance, we can remove over 95% of hallucinations at generation time while retaining 96% of in-support samples. We conclude our exploration by showing the implications of such hallucination (and its removal) on the collapse (and stabilization) of recursive training on synthetic data with experiments on MNIST and 2D Gaussians dataset. We release our code at https://github.com/locuslab/diffusion-model-hallucination.

Diffusion models have demonstrated remarkable capabilities in visual content generation but remain challenging to deploy due to their high computational cost during inference. This computational burden primarily arises from the quadratic complexity of self-attention with respect to image or video resolution. While existing acceleration methods often compromise output quality or necessitate costly retraining, we observe that most diffusion models are pre-trained at lower resolutions, presenting an opportunity to exploit these low-resolution priors for more efficient inference without degrading performance. In this work, we introduce Bottleneck Sampling, a training-free framework that leverages low-resolution priors to reduce computational overhead while preserving output fidelity. Bottleneck Sampling follows a high-low-high denoising workflow: it performs high-resolution denoising in the initial and final stages while operating at lower resolutions in intermediate steps. To mitigate aliasing and blurring artifacts, we further refine the resolution transition points and adaptively shift the denoising timesteps at each stage. We evaluate Bottleneck Sampling on both image and video generation tasks, where extensive experiments demonstrate that it accelerates inference by up to 3times for image generation and 2.5times for video generation, all while maintaining output quality comparable to the standard full-resolution sampling process across multiple evaluation metrics. Code is available at: https://github.com/tyfeld/Bottleneck-Sampling

Diffusion models have achieved remarkable success in imaging inverse problems owing to their powerful generative capabilities. However, existing approaches typically rely on models trained for specific degradation types, limiting their generalizability to various degradation scenarios. To address this limitation, we propose a zero-shot framework capable of handling various imaging inverse problems without model retraining. We introduce a likelihood-guided noise refinement mechanism that derives a closed-form approximation of the likelihood score, simplifying score estimation and avoiding expensive gradient computations. This estimated score is subsequently utilized to refine the model-predicted noise, thereby better aligning the restoration process with the generative framework of diffusion models. In addition, we integrate the Denoising Diffusion Implicit Models (DDIM) sampling strategy to further improve inference efficiency. The proposed mechanism can be applied to both optimization-based and sampling-based schemes, providing an effective and flexible zero-shot solution for imaging inverse problems. Extensive experiments demonstrate that our method achieves superior performance across multiple inverse problems, particularly in compressive sensing, delivering high-quality reconstructions even at an extremely low sampling rate (5%).

Recent studies on inverse problems have proposed posterior samplers that leverage the pre-trained diffusion models as powerful priors. These attempts have paved the way for using diffusion models in a wide range of inverse problems. However, the existing methods entail computationally demanding iterative sampling procedures and optimize a separate solution for each measurement, which leads to limited scalability and lack of generalization capability across unseen samples. To address these limitations, we propose a novel approach, Diffusion prior-based Amortized Variational Inference (DAVI) that solves inverse problems with a diffusion prior from an amortized variational inference perspective. Specifically, instead of separate measurement-wise optimization, our amortized inference learns a function that directly maps measurements to the implicit posterior distributions of corresponding clean data, enabling a single-step posterior sampling even for unseen measurements. Extensive experiments on image restoration tasks, e.g., Gaussian deblur, 4times super-resolution, and box inpainting with two benchmark datasets, demonstrate our approach's superior performance over strong baselines. Code is available at https://github.com/mlvlab/DAVI.

Solving ill-posed inverse problems requires careful formulation of prior beliefs over the signals of interest and an accurate description of their manifestation into noisy measurements. Handcrafted signal priors based on e.g. sparsity are increasingly replaced by data-driven deep generative models, and several groups have recently shown that state-of-the-art score-based diffusion models yield particularly strong performance and flexibility. In this paper, we show that the powerful paradigm of posterior sampling with diffusion models can be extended to include rich, structured, noise models. To that end, we propose a joint conditional reverse diffusion process with learned scores for the noise and signal-generating distribution. We demonstrate strong performance gains across various inverse problems with structured noise, outperforming competitive baselines that use normalizing flows and adversarial networks. This opens up new opportunities and relevant practical applications of diffusion modeling for inverse problems in the context of non-Gaussian measurement models.

Diffusion models have achieved remarkable performance in generative modeling, yet their theoretical foundations are often intricate, and the gap between mathematical formulations in papers and practical open-source implementations can be difficult to bridge. Existing tutorials primarily focus on deriving equations, offering limited guidance on how diffusion models actually operate in code. To address this, we present a concise implementation of approximately 300 lines that explains diffusion models from a code-execution perspective. Our minimal example preserves the essential components -- including forward diffusion, reverse sampling, the noise-prediction network, and the training loop -- while removing unnecessary engineering details. This technical report aims to provide researchers with a clear, implementation-first understanding of how diffusion models work in practice and how code and theory correspond. Our code and pre-trained models are available at: https://github.com/disanda/GM/tree/main/DDPM-DDIM-ClassifierFree.

Diffusion models suffer from slow sample generation at inference time. Therefore, developing a principled framework for fast deterministic/stochastic sampling for a broader class of diffusion models is a promising direction. We propose two complementary frameworks for accelerating sample generation in pre-trained models: Conjugate Integrators and Splitting Integrators. Conjugate integrators generalize DDIM, mapping the reverse diffusion dynamics to a more amenable space for sampling. In contrast, splitting-based integrators, commonly used in molecular dynamics, reduce the numerical simulation error by cleverly alternating between numerical updates involving the data and auxiliary variables. After extensively studying these methods empirically and theoretically, we present a hybrid method that leads to the best-reported performance for diffusion models in augmented spaces. Applied to Phase Space Langevin Diffusion [Pandey & Mandt, 2023] on CIFAR-10, our deterministic and stochastic samplers achieve FID scores of 2.11 and 2.36 in only 100 network function evaluations (NFE) as compared to 2.57 and 2.63 for the best-performing baselines, respectively. Our code and model checkpoints will be made publicly available at https://github.com/mandt-lab/PSLD.

Diffusion models are powerful generative models but suffer from slow sampling, often taking 1000 sequential denoising steps for one sample. As a result, considerable efforts have been directed toward reducing the number of denoising steps, but these methods hurt sample quality. Instead of reducing the number of denoising steps (trading quality for speed), in this paper we explore an orthogonal approach: can we run the denoising steps in parallel (trading compute for speed)? In spite of the sequential nature of the denoising steps, we show that surprisingly it is possible to parallelize sampling via Picard iterations, by guessing the solution of future denoising steps and iteratively refining until convergence. With this insight, we present ParaDiGMS, a novel method to accelerate the sampling of pretrained diffusion models by denoising multiple steps in parallel. ParaDiGMS is the first diffusion sampling method that enables trading compute for speed and is even compatible with existing fast sampling techniques such as DDIM and DPMSolver. Using ParaDiGMS, we improve sampling speed by 2-4x across a range of robotics and image generation models, giving state-of-the-art sampling speeds of 0.2s on 100-step DiffusionPolicy and 16s on 1000-step StableDiffusion-v2 with no measurable degradation of task reward, FID score, or CLIP score.

One of the key components within diffusion models is the UNet for noise prediction. While several works have explored basic properties of the UNet decoder, its encoder largely remains unexplored. In this work, we conduct the first comprehensive study of the UNet encoder. We empirically analyze the encoder features and provide insights to important questions regarding their changes at the inference process. In particular, we find that encoder features change gently, whereas the decoder features exhibit substantial variations across different time-steps. This finding inspired us to omit the encoder at certain adjacent time-steps and reuse cyclically the encoder features in the previous time-steps for the decoder. Further based on this observation, we introduce a simple yet effective encoder propagation scheme to accelerate the diffusion sampling for a diverse set of tasks. By benefiting from our propagation scheme, we are able to perform in parallel the decoder at certain adjacent time-steps. Additionally, we introduce a prior noise injection method to improve the texture details in the generated image. Besides the standard text-to-image task, we also validate our approach on other tasks: text-to-video, personalized generation and reference-guided generation. Without utilizing any knowledge distillation technique, our approach accelerates both the Stable Diffusion (SD) and the DeepFloyd-IF models sampling by 41% and 24% respectively, while maintaining high-quality generation performance. Our code is available in https://github.com/hutaiHang/Faster-Diffusion{FasterDiffusion}.

Diffusion-based samplers learn to sample complex, high-dimensional distributions using energies or log densities alone, without training data. Yet, they remain impractical for molecular sampling because they are often slower than molecular dynamics and miss thermodynamically relevant modes. Inspired by enhanced sampling, we encourage exploration by introducing a sequential bias along bespoke, information-rich, low-dimensional projections of atomic coordinates known as collective variables (CVs). We introduce a repulsive potential centered on the CVs from recent samples, which pushes future samples towards novel CV regions and effectively increases the temperature in the projected space. Our resulting method improves efficiency, mode discovery, enables the estimation of free energy differences, and retains independent sampling from the approximate Boltzmann distribution via reweighting by the bias. On standard peptide conformational sampling benchmarks, the method recovers diverse conformational states and accurate free energy profiles. We are the first to demonstrate reactive sampling using a diffusion-based sampler, capturing bond breaking and formation with universal interatomic potentials at near-first-principles accuracy. The approach resolves reactive energy landscapes at a fraction of the wall-clock time of standard sampling methods, advancing diffusion-based sampling towards practical use in molecular sciences.

Diffusion models, which learn to reverse a signal destruction process to generate new data, typically require the signal at each step to have the same dimension. We argue that, considering the spatial redundancy in image signals, there is no need to maintain a high dimensionality in the evolution process, especially in the early generation phase. To this end, we make a theoretical generalization of the forward diffusion process via signal decomposition. Concretely, we manage to decompose an image into multiple orthogonal components and control the attenuation of each component when perturbing the image. That way, along with the noise strength increasing, we are able to diminish those inconsequential components and thus use a lower-dimensional signal to represent the source, barely losing information. Such a reformulation allows to vary dimensions in both training and inference of diffusion models. Extensive experiments on a range of datasets suggest that our approach substantially reduces the computational cost and achieves on-par or even better synthesis performance compared to baseline methods. We also show that our strategy facilitates high-resolution image synthesis and improves FID of diffusion model trained on FFHQ at 1024times1024 resolution from 52.40 to 10.46. Code and models will be made publicly available.

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate O(1/{T}^2) with T the number of steps, improving upon the O(1/T) rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate O(1/T), outperforming the rate O(1/T) for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates ell_2-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.

Flow maps enable high-quality image generation in a single forward pass. However, unlike iterative diffusion models, their lack of an explicit sampling trajectory impedes incorporating external constraints for conditional generation and solving inverse problems. We put forth Variational Flow Maps, a framework for conditional sampling that shifts the perspective of conditioning from "guiding a sampling path", to that of "learning the proper initial noise". Specifically, given an observation, we seek to learn a noise adapter model that outputs a noise distribution, so that after mapping to the data space via flow map, the samples respect the observation and data prior. To this end, we develop a principled variational objective that jointly trains the noise adapter and the flow map, improving noise-data alignment, such that sampling from complex data posterior is achieved with a simple adapter. Experiments on various inverse problems show that VFMs produce well-calibrated conditional samples in a single (or few) steps. For ImageNet, VFM attains competitive fidelity while accelerating the sampling by orders of magnitude compared to alternative iterative diffusion/flow models. Code is available at https://github.com/abbasmammadov/VFM

Diffusion-based image restoration (IR) methods aim to use diffusion models to recover high-quality (HQ) images from degraded images and achieve promising performance. Due to the inherent property of diffusion models, most of these methods need long serial sampling chains to restore HQ images step-by-step. As a result, it leads to expensive sampling time and high computation costs. Moreover, such long sampling chains hinder understanding the relationship between the restoration results and the inputs since it is hard to compute the gradients in the whole chains. In this work, we aim to rethink the diffusion-based IR models through a different perspective, i.e., a deep equilibrium (DEQ) fixed point system. Specifically, we derive an analytical solution by modeling the entire sampling chain in diffusion-based IR models as a joint multivariate fixed point system. With the help of the analytical solution, we are able to conduct single-image sampling in a parallel way and restore HQ images without training. Furthermore, we compute fast gradients in DEQ and found that initialization optimization can boost performance and control the generation direction. Extensive experiments on benchmarks demonstrate the effectiveness of our proposed method on typical IR tasks and real-world settings. The code and models will be made publicly available.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

Current diffusion-based image restoration methods feed degraded input images as conditions into the noise estimation network. However, interpreting this diffusion process is challenging since it essentially generates the target image from the noise. To establish a unified and more interpretable model for image generation and restoration, we propose residual denoising diffusion models (RDDM). In contrast to existing diffusion models (e.g., DDPM or DDIM) that focus solely on noise estimation, our RDDM predicts residuals to represent directional diffusion from the target domain to the input domain, while concurrently estimating noise to account for random perturbations in the diffusion process. The introduction of residuals allows us to redefine the forward diffusion process, wherein the target image progressively diffuses into a purely noisy image or a noise-carrying input image, thus unifying image generation and restoration. We demonstrate that our sampling process is consistent with that of DDPM and DDIM through coefficient transformation, and propose a partially path-independent generation process to better understand the reverse process. Notably, with native support for conditional inputs, our RDDM enables a generic UNet, trained with only an ell _1 loss and a batch size of 1, to compete with state-of-the-art image restoration methods. We provide code and pre-trained models to encourage further exploration, application, and development of our innovative framework (https://github.com/nachifur/RDDM).

We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at https://github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.

Diffusion probabilistic models (DPMs) have shown remarkable performance in visual synthesis but are computationally expensive due to the need for multiple evaluations during the sampling. Recent predictor-corrector diffusion samplers have significantly reduced the required number of function evaluations (NFE), but inherently suffer from a misalignment issue caused by the extra corrector step, especially with a large classifier-free guidance scale (CFG). In this paper, we introduce a new fast DPM sampler called DC-Solver, which leverages dynamic compensation (DC) to mitigate the misalignment of the predictor-corrector samplers. The dynamic compensation is controlled by compensation ratios that are adaptive to the sampling steps and can be optimized on only 10 datapoints by pushing the sampling trajectory toward a ground truth trajectory. We further propose a cascade polynomial regression (CPR) which can instantly predict the compensation ratios on unseen sampling configurations. Additionally, we find that the proposed dynamic compensation can also serve as a plug-and-play module to boost the performance of predictor-only samplers. Extensive experiments on both unconditional sampling and conditional sampling demonstrate that our DC-Solver can consistently improve the sampling quality over previous methods on different DPMs with a wide range of resolutions up to 1024times1024. Notably, we achieve 10.38 FID (NFE=5) on unconditional FFHQ and 0.394 MSE (NFE=5, CFG=7.5) on Stable-Diffusion-2.1. Code is available at https://github.com/wl-zhao/DC-Solver

Text-conditioned image generation has made significant progress in recent years with generative adversarial networks and more recently, diffusion models. While diffusion models conditioned on text prompts have produced impressive and high-quality images, accurately representing complex text prompts such as the number of instances of a specific object remains challenging. To address this limitation, we propose a novel guidance approach for the sampling process in the diffusion model that leverages bounding box and segmentation map information at inference time without additional training data. Through a novel loss in the sampling process, our approach guides the model with semantic features from CLIP embeddings and enforces geometric constraints, leading to high-resolution images that accurately represent the scene. To obtain bounding box and segmentation map information, we structure the text prompt as a scene graph and enrich the nodes with CLIP embeddings. Our proposed model achieves state-of-the-art performance on two public benchmarks for image generation from scene graphs, surpassing both scene graph to image and text-based diffusion models in various metrics. Our results demonstrate the effectiveness of incorporating bounding box and segmentation map guidance in the diffusion model sampling process for more accurate text-to-image generation.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

Conditional generative models typically demand large annotated training sets to achieve high-quality synthesis. As a result, there has been significant interest in designing models that perform plug-and-play generation, i.e., to use a predefined or pretrained model, which is not explicitly trained on the generative task, to guide the generative process (e.g., using language). However, such guidance is typically useful only towards synthesizing high-level semantics rather than editing fine-grained details as in image-to-image translation tasks. To this end, and capitalizing on the powerful fine-grained generative control offered by the recent diffusion-based generative models, we introduce Steered Diffusion, a generalized framework for photorealistic zero-shot conditional image generation using a diffusion model trained for unconditional generation. The key idea is to steer the image generation of the diffusion model at inference time via designing a loss using a pre-trained inverse model that characterizes the conditional task. This loss modulates the sampling trajectory of the diffusion process. Our framework allows for easy incorporation of multiple conditions during inference. We present experiments using steered diffusion on several tasks including inpainting, colorization, text-guided semantic editing, and image super-resolution. Our results demonstrate clear qualitative and quantitative improvements over state-of-the-art diffusion-based plug-and-play models while adding negligible additional computational cost.

Subsampling is commonly used to mitigate costs associated with data acquisition, such as time or energy requirements, motivating the development of algorithms for estimating the fully-sampled signal of interest x from partially observed measurements y. In maximum-entropy sampling, one selects measurement locations that are expected to have the highest entropy, so as to minimize uncertainty about x. This approach relies on an accurate model of the posterior distribution over future measurements, given the measurements observed so far. Recently, diffusion models have been shown to produce high-quality posterior samples of high-dimensional signals using guided diffusion. In this work, we propose Active Diffusion Subsampling (ADS), a method for performing active subsampling using guided diffusion in which the model tracks a distribution of beliefs over the true state of x throughout the reverse diffusion process, progressively decreasing its uncertainty by choosing to acquire measurements with maximum expected entropy, and ultimately generating the posterior distribution p(x | y). ADS can be applied using pre-trained diffusion models for any subsampling rate, and does not require task-specific retraining - just the specification of a measurement model. Furthermore, the maximum entropy sampling policy employed by ADS is interpretable, enhancing transparency relative to existing methods using black-box policies. Experimentally, we show that ADS outperforms fixed sampling strategies, and study an application of ADS in Magnetic Resonance Imaging acceleration using the fastMRI dataset, finding that ADS performs competitively with supervised methods. Code available at https://active-diffusion-subsampling.github.io/.

Diffusion models excel at capturing the natural design spaces of images, molecules, DNA, RNA, and protein sequences. However, rather than merely generating designs that are natural, we often aim to optimize downstream reward functions while preserving the naturalness of these design spaces. Existing methods for achieving this goal often require ``differentiable'' proxy models (e.g., classifier guidance or DPS) or involve computationally expensive fine-tuning of diffusion models (e.g., classifier-free guidance, RL-based fine-tuning). In our work, we propose a new method to address these challenges. Our algorithm is an iterative sampling method that integrates soft value functions, which looks ahead to how intermediate noisy states lead to high rewards in the future, into the standard inference procedure of pre-trained diffusion models. Notably, our approach avoids fine-tuning generative models and eliminates the need to construct differentiable models. This enables us to (1) directly utilize non-differentiable features/reward feedback, commonly used in many scientific domains, and (2) apply our method to recent discrete diffusion models in a principled way. Finally, we demonstrate the effectiveness of our algorithm across several domains, including image generation, molecule generation, and DNA/RNA sequence generation. The code is available at https://github.com/masa-ue/SVDD{https://github.com/masa-ue/SVDD}.

Although text-to-image diffusion models have made significant strides in generating images from text, they are sometimes more inclined to generate images like the data on which the model was trained rather than the provided text. This limitation has hindered their usage in both 2D and 3D applications. To address this problem, we explored the use of negative prompts but found that the current implementation fails to produce desired results, particularly when there is an overlap between the main and negative prompts. To overcome this issue, we propose Perp-Neg, a new algorithm that leverages the geometrical properties of the score space to address the shortcomings of the current negative prompts algorithm. Perp-Neg does not require any training or fine-tuning of the model. Moreover, we experimentally demonstrate that Perp-Neg provides greater flexibility in generating images by enabling users to edit out unwanted concepts from the initially generated images in 2D cases. Furthermore, to extend the application of Perp-Neg to 3D, we conducted a thorough exploration of how Perp-Neg can be used in 2D to condition the diffusion model to generate desired views, rather than being biased toward the canonical views. Finally, we applied our 2D intuition to integrate Perp-Neg with the state-of-the-art text-to-3D (DreamFusion) method, effectively addressing its Janus (multi-head) problem. Our project page is available at https://Perp-Neg.github.io/

Recent years have witnessed the rapid progress and broad application of diffusion probabilistic models (DPMs). Sampling from DPMs can be viewed as solving an ordinary differential equation (ODE). Despite the promising performance, the generation of DPMs usually consumes much time due to the large number of function evaluations (NFE). Though recent works have accelerated the sampling to around 20 steps with high-order solvers, the sample quality with less than 10 NFE can still be improved. In this paper, we propose a unified sampling framework (USF) to study the optional strategies for solver. Under this framework, we further reveal that taking different solving strategies at different timesteps may help further decrease the truncation error, and a carefully designed solver schedule has the potential to improve the sample quality by a large margin. Therefore, we propose a new sampling framework based on the exponential integral formulation that allows free choices of solver strategy at each step and design specific decisions for the framework. Moreover, we propose S^3, a predictor-based search method that automatically optimizes the solver schedule to get a better time-quality trade-off of sampling. We demonstrate that S^3 can find outstanding solver schedules which outperform the state-of-the-art sampling methods on CIFAR-10, CelebA, ImageNet, and LSUN-Bedroom datasets. Specifically, we achieve 2.69 FID with 10 NFE and 6.86 FID with 5 NFE on CIFAR-10 dataset, outperforming the SOTA method significantly. We further apply S^3 to Stable-Diffusion model and get an acceleration ratio of 2times, showing the feasibility of sampling in very few steps without retraining the neural network.

We propose a new method for solving imaging inverse problems using text-to-image latent diffusion models as general priors. Existing methods using latent diffusion models for inverse problems typically rely on simple null text prompts, which can lead to suboptimal performance. To address this limitation, we introduce a method for prompt tuning, which jointly optimizes the text embedding on-the-fly while running the reverse diffusion process. This allows us to generate images that are more faithful to the diffusion prior. In addition, we propose a method to keep the evolution of latent variables within the range space of the encoder, by projection. This helps to reduce image artifacts, a major problem when using latent diffusion models instead of pixel-based diffusion models. Our combined method, called P2L, outperforms both image- and latent-diffusion model-based inverse problem solvers on a variety of tasks, such as super-resolution, deblurring, and inpainting.

Diffusion models have been successful on a range of conditional generation tasks including molecular design and text-to-image generation. However, these achievements have primarily depended on task-specific conditional training or error-prone heuristic approximations. Ideally, a conditional generation method should provide exact samples for a broad range of conditional distributions without requiring task-specific training. To this end, we introduce the Twisted Diffusion Sampler, or TDS. TDS is a sequential Monte Carlo (SMC) algorithm that targets the conditional distributions of diffusion models through simulating a set of weighted particles. The main idea is to use twisting, an SMC technique that enjoys good computational efficiency, to incorporate heuristic approximations without compromising asymptotic exactness. We first find in simulation and in conditional image generation tasks that TDS provides a computational statistical trade-off, yielding more accurate approximations with many particles but with empirical improvements over heuristics with as few as two particles. We then turn to motif-scaffolding, a core task in protein design, using a TDS extension to Riemannian diffusion models. On benchmark test cases, TDS allows flexible conditioning criteria and often outperforms the state of the art.

With the rapid development of diffusion models and flow-based generative models, there has been a surge of interests in solving noisy linear inverse problems, e.g., super-resolution, deblurring, denoising, colorization, etc, with generative models. However, while remarkable reconstruction performances have been achieved, their inference time is typically too slow since most of them rely on the seminal diffusion posterior sampling (DPS) framework and thus to approximate the intractable likelihood score, time-consuming gradient calculation through back-propagation is needed. To address this issue, this paper provides a fast and effective solution by proposing a simple closed-form approximation to the likelihood score. For both diffusion and flow-based models, extensive experiments are conducted on various noisy linear inverse problems such as noisy super-resolution, denoising, deblurring, and colorization. In all these tasks, our method (namely DMPS) demonstrates highly competitive or even better reconstruction performances while being significantly faster than all the baseline methods.

Diffusion models are a family of generative models that yield record-breaking performance in tasks such as image synthesis, video generation, and molecule design. Despite their capabilities, their efficiency, especially in the reverse denoising process, remains a challenge due to slow convergence rates and high computational costs. In this work, we introduce an approach that leverages continuous dynamical systems to design a novel denoising network for diffusion models that is more parameter-efficient, exhibits faster convergence, and demonstrates increased noise robustness. Experimenting with denoising probabilistic diffusion models, our framework operates with approximately a quarter of the parameters and 30% of the Floating Point Operations (FLOPs) compared to standard U-Nets in Denoising Diffusion Probabilistic Models (DDPMs). Furthermore, our model is up to 70% faster in inference than the baseline models when measured in equal conditions while converging to better quality solutions.

We present a novel generative approach based on Denoising Diffusion Models (DDMs), which produces high-quality image samples along with their losslessly compressed bit-stream representations. This is obtained by replacing the standard Gaussian noise sampling in the reverse diffusion with a selection of noise samples from pre-defined codebooks of fixed iid Gaussian vectors. Surprisingly, we find that our method, termed Denoising Diffusion Codebook Model (DDCM), retains sample quality and diversity of standard DDMs, even for extremely small codebooks. We leverage DDCM and pick the noises from the codebooks that best match a given image, converting our generative model into a highly effective lossy image codec achieving state-of-the-art perceptual image compression results. More generally, by setting other noise selections rules, we extend our compression method to any conditional image generation task (e.g., image restoration), where the generated images are produced jointly with their condensed bit-stream representations. Our work is accompanied by a mathematical interpretation of the proposed compressed conditional generation schemes, establishing a connection with score-based approximations of posterior samplers for the tasks considered.

Uniform-state discrete diffusion models excel at few-step generation and guidance due to their ability to self-correct, making them preferred over autoregressive or Masked diffusion models in these settings. However, their sampling quality plateaus with ancestral samplers as the number of steps increases. We introduce a family of Predictor-Corrector (PC) samplers for discrete diffusion that generalize prior methods and apply to arbitrary noise processes. When paired with uniform-state diffusion, our samplers outperform ancestral sampling on both language and image modeling, achieving lower generative perplexity at matched unigram entropy on OpenWebText and better FID/IS scores on CIFAR10. Crucially, unlike conventional samplers, our PC methods continue to improve with more sampling steps. Taken together, these findings call into question the assumption that Masked diffusion is the inevitable future of diffusion-based language modeling. Beyond sampling, we develop a memory-efficient curriculum for the Gaussian relaxation training phase, reducing training time by 25% and memory by 33% compared to Duo while maintaining comparable perplexity on OpenWebText and LM1B and strong downstream performance. We release code, checkpoints, and a video-tutorial on: https://s-sahoo.com/duo-ch2

In this paper we consider the problem of acoustic inversion in the context of the optoacoustic tomography image reconstruction problem. By leveraging the ability of the recently proposed diffusion models for image generative tasks among others, we devise an image reconstruction architecture based on a conditional diffusion process. The scheme makes use of an initial image reconstruction, which is preprocessed by an autoencoder to generate an adequate representation. This representation is used as conditional information in a generative diffusion process. Although the computational requirements for training and implementing the architecture are not low, several design choices discussed in the work were made to keep them manageable. Numerical results show that the conditional information allows to properly bias the parameters of the diffusion model to improve the quality of the initial reconstructed image, eliminating artifacts or even reconstructing finer details of the ground-truth image that are not recoverable by the initial image reconstruction method. We also tested the proposal under experimental conditions and the obtained results were in line with those corresponding to the numerical simulations. Improvements in image quality up to 17 % in terms of peak signal-to-noise ratio were observed.

Diffusion models have demonstrated compelling generation quality by optimizing the variational lower bound through a simple denoising score matching loss. In this paper, we provide theoretical evidence that the prevailing practice of using a constant loss weight strategy in diffusion models leads to biased estimation during the training phase. Simply optimizing the denoising network to predict Gaussian noise with constant weighting may hinder precise estimations of original images. To address the issue, we propose an elegant and effective weighting strategy grounded in the theoretically unbiased principle. Moreover, we conduct a comprehensive and systematic exploration to dissect the inherent bias problem deriving from constant weighting loss from the perspectives of its existence, impact and reasons. These analyses are expected to advance our understanding and demystify the inner workings of diffusion models. Through empirical evaluation, we demonstrate that our proposed debiased estimation method significantly enhances sample quality without the reliance on complex techniques, and exhibits improved efficiency compared to the baseline method both in training and sampling processes.

Diffusion-based generative models (DBGMs) perturb data to a target noise distribution and reverse this process to generate samples. The choice of noising process, or inference diffusion process, affects both likelihoods and sample quality. For example, extending the inference process with auxiliary variables leads to improved sample quality. While there are many such multivariate diffusions to explore, each new one requires significant model-specific analysis, hindering rapid prototyping and evaluation. In this work, we study Multivariate Diffusion Models (MDMs). For any number of auxiliary variables, we provide a recipe for maximizing a lower-bound on the MDMs likelihood without requiring any model-specific analysis. We then demonstrate how to parameterize the diffusion for a specified target noise distribution; these two points together enable optimizing the inference diffusion process. Optimizing the diffusion expands easy experimentation from just a few well-known processes to an automatic search over all linear diffusions. To demonstrate these ideas, we introduce two new specific diffusions as well as learn a diffusion process on the MNIST, CIFAR10, and ImageNet32 datasets. We show learned MDMs match or surpass bits-per-dims (BPDs) relative to fixed choices of diffusions for a given dataset and model architecture.

Score-based diffusion modeling is a generative machine learning algorithm that can be used to sample from complex distributions. They achieve this by learning a score function, i.e., the gradient of the log-probability density of the data, and reversing a noising process using the same. Once trained, score-based diffusion models not only generate new samples but also enable zero-shot conditioning of the generated samples on observed data. This promises a novel paradigm for data and model fusion, wherein the implicitly learned distributions of pretrained score-based diffusion models can be updated given the availability of online data in a Bayesian formulation. In this article, we apply such a concept to the super-resolution of a high-dimensional dynamical system, given the real-time availability of low-resolution and experimentally observed sparse sensor measurements from multimodal data. Additional analysis on how score-based sampling can be used for uncertainty estimates is also provided. Our experiments are performed for a super-resolution task that generates the ERA5 atmospheric dataset given sparse observations from a coarse-grained representation of the same and/or from unstructured experimental observations of the IGRA radiosonde dataset. We demonstrate accurate recovery of the high dimensional state given multiple sources of low-fidelity measurements. We also discover that the generative model can balance the influence of multiple dataset modalities during spatiotemporal reconstructions.

Recent advancements in text-to-image diffusion models have brought them to the public spotlight, becoming widely accessible and embraced by everyday users. However, these models have been shown to generate harmful content such as not-safe-for-work (NSFW) images. While approaches have been proposed to erase such abstract concepts from the models, jail-breaking techniques have succeeded in bypassing such safety measures. In this paper, we propose TraSCE, an approach to guide the diffusion trajectory away from generating harmful content. Our approach is based on negative prompting, but as we show in this paper, a widely used negative prompting strategy is not a complete solution and can easily be bypassed in some corner cases. To address this issue, we first propose using a specific formulation of negative prompting instead of the widely used one. Furthermore, we introduce a localized loss-based guidance that enhances the modified negative prompting technique by steering the diffusion trajectory. We demonstrate that our proposed method achieves state-of-the-art results on various benchmarks in removing harmful content, including ones proposed by red teams, and erasing artistic styles and objects. Our proposed approach does not require any training, weight modifications, or training data (either image or prompt), making it easier for model owners to erase new concepts.

Diffusion models generating images conditionally on text, such as Dall-E 2 and Stable Diffusion, have recently made a splash far beyond the computer vision community. Here, we tackle the related problem of generating point clouds, both unconditionally, and conditionally with images. For the latter, we introduce a novel geometrically-motivated conditioning scheme based on projecting sparse image features into the point cloud and attaching them to each individual point, at every step in the denoising process. This approach improves geometric consistency and yields greater fidelity than current methods relying on unstructured, global latent codes. Additionally, we show how to apply recent continuous-time diffusion schemes. Our method performs on par or above the state of art on conditional and unconditional experiments on synthetic data, while being faster, lighter, and delivering tractable likelihoods. We show it can also scale to diverse indoors scenes.

We present the first framework to solve linear inverse problems leveraging pre-trained latent diffusion models. Previously proposed algorithms (such as DPS and DDRM) only apply to pixel-space diffusion models. We theoretically analyze our algorithm showing provable sample recovery in a linear model setting. The algorithmic insight obtained from our analysis extends to more general settings often considered in practice. Experimentally, we outperform previously proposed posterior sampling algorithms in a wide variety of problems including random inpainting, block inpainting, denoising, deblurring, destriping, and super-resolution.

Sampling from the posterior distribution poses a major computational challenge in solving inverse problems using latent diffusion models. Common methods rely on Tweedie's first-order moments, which are known to induce a quality-limiting bias. Existing second-order approximations are impractical due to prohibitive computational costs, making standard reverse diffusion processes intractable for posterior sampling. This paper introduces Second-order Tweedie sampler from Surrogate Loss (STSL), a novel sampler that offers efficiency comparable to first-order Tweedie with a tractable reverse process using second-order approximation. Our theoretical results reveal that the second-order approximation is lower bounded by our surrogate loss that only requires O(1) compute using the trace of the Hessian, and by the lower bound we derive a new drift term to make the reverse process tractable. Our method surpasses SoTA solvers PSLD and P2L, achieving 4X and 8X reduction in neural function evaluations, respectively, while notably enhancing sampling quality on FFHQ, ImageNet, and COCO benchmarks. In addition, we show STSL extends to text-guided image editing and addresses residual distortions present from corrupted images in leading text-guided image editing methods. To our best knowledge, this is the first work to offer an efficient second-order approximation in solving inverse problems using latent diffusion and editing real-world images with corruptions.

We introduce infty-Diff, a generative diffusion model which directly operates on infinite resolution data. By randomly sampling subsets of coordinates during training and learning to denoise the content at those coordinates, a continuous function is learned that allows sampling at arbitrary resolutions. In contrast to other recent infinite resolution generative models, our approach operates directly on the raw data, not requiring latent vector compression for context, using hypernetworks, nor relying on discrete components. As such, our approach achieves significantly higher sample quality, as evidenced by lower FID scores, as well as being able to effectively scale to higher resolutions than the training data while retaining detail.

Recently, diffusion probabilistic models (DPMs) have achieved promising results in diverse generative tasks. A typical DPM framework includes a forward process that gradually diffuses the data distribution and a reverse process that recovers the data distribution from time-dependent data scores. In this work, we observe that the stochastic reverse process of data scores is a martingale, from which concentration bounds and the optional stopping theorem for data scores can be derived. Then, we discover a simple way for calibrating an arbitrary pretrained DPM, with which the score matching loss can be reduced and the lower bounds of model likelihood can consequently be increased. We provide general calibration guidelines under various model parametrizations. Our calibration method is performed only once and the resulting models can be used repeatedly for sampling. We conduct experiments on multiple datasets to empirically validate our proposal. Our code is at https://github.com/thudzj/Calibrated-DPMs.

This study presents a new image super-resolution (SR) technique based on diffusion inversion, aiming at harnessing the rich image priors encapsulated in large pre-trained diffusion models to improve SR performance. We design a Partial noise Prediction strategy to construct an intermediate state of the diffusion model, which serves as the starting sampling point. Central to our approach is a deep noise predictor to estimate the optimal noise maps for the forward diffusion process. Once trained, this noise predictor can be used to initialize the sampling process partially along the diffusion trajectory, generating the desirable high-resolution result. Compared to existing approaches, our method offers a flexible and efficient sampling mechanism that supports an arbitrary number of sampling steps, ranging from one to five. Even with a single sampling step, our method demonstrates superior or comparable performance to recent state-of-the-art approaches. The code and model are publicly available at https://github.com/zsyOAOA/InvSR.

Imputation of missing images via source-to-target modality translation can improve diversity in medical imaging protocols. A pervasive approach for synthesizing target images involves one-shot mapping through generative adversarial networks (GAN). Yet, GAN models that implicitly characterize the image distribution can suffer from limited sample fidelity. Here, we propose a novel method based on adversarial diffusion modeling, SynDiff, for improved performance in medical image translation. To capture a direct correlate of the image distribution, SynDiff leverages a conditional diffusion process that progressively maps noise and source images onto the target image. For fast and accurate image sampling during inference, large diffusion steps are taken with adversarial projections in the reverse diffusion direction. To enable training on unpaired datasets, a cycle-consistent architecture is devised with coupled diffusive and non-diffusive modules that bilaterally translate between two modalities. Extensive assessments are reported on the utility of SynDiff against competing GAN and diffusion models in multi-contrast MRI and MRI-CT translation. Our demonstrations indicate that SynDiff offers quantitatively and qualitatively superior performance against competing baselines.

While diffusion-based image restoration (IR) methods have achieved remarkable success, they are still limited by the low inference speed attributed to the necessity of executing hundreds or even thousands of sampling steps. Existing acceleration sampling techniques, though seeking to expedite the process, inevitably sacrifice performance to some extent, resulting in over-blurry restored outcomes. To address this issue, this study proposes a novel and efficient diffusion model for IR that significantly reduces the required number of diffusion steps. Our method avoids the need for post-acceleration during inference, thereby avoiding the associated performance deterioration. Specifically, our proposed method establishes a Markov chain that facilitates the transitions between the high-quality and low-quality images by shifting their residuals, substantially improving the transition efficiency. A carefully formulated noise schedule is devised to flexibly control the shifting speed and the noise strength during the diffusion process. Extensive experimental evaluations demonstrate that the proposed method achieves superior or comparable performance to current state-of-the-art methods on three classical IR tasks, namely image super-resolution, image inpainting, and blind face restoration, \textbf{even only with four sampling steps}. Our code and model are publicly available at https://github.com/zsyOAOA/ResShift.

Diffusion models are a powerful generative framework, but come with expensive inference. Existing acceleration methods often compromise image quality or fail under complex conditioning when operating in an extremely low-step regime. In this work, we propose a novel distillation framework tailored to enable high-fidelity, diverse sample generation using just one to three steps. Our approach comprises three key components: (i) Backward Distillation, which mitigates training-inference discrepancies by calibrating the student on its own backward trajectory; (ii) Shifted Reconstruction Loss that dynamically adapts knowledge transfer based on the current time step; and (iii) Noise Correction, an inference-time technique that enhances sample quality by addressing singularities in noise prediction. Through extensive experiments, we demonstrate that our method outperforms existing competitors in quantitative metrics and human evaluations. Remarkably, it achieves performance comparable to the teacher model using only three denoising steps, enabling efficient high-quality generation.

Diffusion probabilistic models (DPMs) have demonstrated a very promising ability in high-resolution image synthesis. However, sampling from a pre-trained DPM usually requires hundreds of model evaluations, which is computationally expensive. Despite recent progress in designing high-order solvers for DPMs, there still exists room for further speedup, especially in extremely few steps (e.g., 5~10 steps). Inspired by the predictor-corrector for ODE solvers, we develop a unified corrector (UniC) that can be applied after any existing DPM sampler to increase the order of accuracy without extra model evaluations, and derive a unified predictor (UniP) that supports arbitrary order as a byproduct. Combining UniP and UniC, we propose a unified predictor-corrector framework called UniPC for the fast sampling of DPMs, which has a unified analytical form for any order and can significantly improve the sampling quality over previous methods. We evaluate our methods through extensive experiments including both unconditional and conditional sampling using pixel-space and latent-space DPMs. Our UniPC can achieve 3.87 FID on CIFAR10 (unconditional) and 7.51 FID on ImageNet 256times256 (conditional) with only 10 function evaluations. Code is available at https://github.com/wl-zhao/UniPC

The probabilistic diffusion model has become highly effective across various domains. Typically, sampling from a diffusion model involves using a denoising distribution characterized by a Gaussian with a learned mean and either fixed or learned covariances. In this paper, we leverage the recently proposed covariance moment matching technique and introduce a novel method for learning the diagonal covariance. Unlike traditional data-driven diagonal covariance approximation approaches, our method involves directly regressing the optimal diagonal analytic covariance using a new, unbiased objective named Optimal Covariance Matching (OCM). This approach can significantly reduce the approximation error in covariance prediction. We demonstrate how our method can substantially enhance the sampling efficiency, recall rate and likelihood of commonly used diffusion models.

Diffusion models have emerged as a powerful new family of deep generative models with record-breaking performance in many applications, including image synthesis, video generation, and molecule design. In this survey, we provide an overview of the rapidly expanding body of work on diffusion models, categorizing the research into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. We also discuss the potential for combining diffusion models with other generative models for enhanced results. We further review the wide-ranging applications of diffusion models in fields spanning from computer vision, natural language generation, temporal data modeling, to interdisciplinary applications in other scientific disciplines. This survey aims to provide a contextualized, in-depth look at the state of diffusion models, identifying the key areas of focus and pointing to potential areas for further exploration. Github: https://github.com/YangLing0818/Diffusion-Models-Papers-Survey-Taxonomy.

In recent years, diffusion models have gained popularity for their ability to generate higher-quality images in comparison to GAN models. However, like any other large generative models, these models require a huge amount of data, computational resources, and meticulous tuning for successful training. This poses a significant challenge, rendering it infeasible for most individuals. As a result, the research community has devised methods to leverage pre-trained unconditional diffusion models with additional guidance for the purpose of conditional image generative. These methods enable conditional image generations on diverse inputs and, most importantly, circumvent the need for training the diffusion model. In this paper, our objective is to reduce the time-required and computational overhead introduced by the addition of guidance in diffusion models -- while maintaining comparable image quality. We propose a set of methods based on our empirical analysis, demonstrating a reduction in computation time by approximately threefold.

Diffusion models, which employ stochastic differential equations to sample images through integrals, have emerged as a dominant class of generative models. However, the rationality of the diffusion process itself receives limited attention, leaving the question of whether the problem is well-posed and well-conditioned. In this paper, we uncover a vexing propensity of diffusion models: they frequently exhibit the infinite Lipschitz near the zero point of timesteps. This poses a threat to the stability and accuracy of the diffusion process, which relies on integral operations. We provide a comprehensive evaluation of the issue from both theoretical and empirical perspectives. To address this challenge, we propose a novel approach, dubbed E-TSDM, which eliminates the Lipschitz singularity of the diffusion model near zero. Remarkably, our technique yields a substantial improvement in performance, e.g., on the high-resolution FFHQ dataset (256times256). Moreover, as a byproduct of our method, we manage to achieve a dramatic reduction in the Frechet Inception Distance of other acceleration methods relying on network Lipschitz, including DDIM and DPM-Solver, by over 33%. We conduct extensive experiments on diverse datasets to validate our theory and method. Our work not only advances the understanding of the general diffusion process, but also provides insights for the design of diffusion models.

Stochastic differential equation (SDE)-based generative models have achieved substantial progress in conditional generation via training-free differentiable loss-guided approaches. However, existing methodologies utilizing posterior sam- pling typically confront a substantial estimation error, which results in inaccu- rate gradients for guidance and leading to inconsistent generation results. To mitigate this issue, we propose that performing an additional backward denois- ing step and Monte-Carlo sampling (ABMS) can achieve better guided diffu- sion, which is a plug-and-play adjustment strategy. To verify the effectiveness of our method, we provide theoretical analysis and propose the adoption of a dual-focus evaluation framework, which further serves to highlight the critical problem of cross-condition interference prevalent in existing approaches. We conduct experiments across various task settings and data types, mainly includ- ing conditional online handwritten trajectory generation, image inverse problems (inpainting, super resolution and gaussian deblurring) molecular inverse design and so on. Experimental results demonstrate that our approach can be effec- tively used with higher order samplers and consistently improves the quality of generation samples across all the different scenarios.

Diffusion models have achieved remarkable success across various domains. However, their slow generation speed remains a critical challenge. Existing acceleration methods, while aiming to reduce steps, often compromise sample quality, controllability, or introduce training complexities. Therefore, we propose RayFlow, a novel diffusion framework that addresses these limitations. Unlike previous methods, RayFlow guides each sample along a unique path towards an instance-specific target distribution. This method minimizes sampling steps while preserving generation diversity and stability. Furthermore, we introduce Time Sampler, an importance sampling technique to enhance training efficiency by focusing on crucial timesteps. Extensive experiments demonstrate RayFlow's superiority in generating high-quality images with improved speed, control, and training efficiency compared to existing acceleration techniques.

Diffusion models have recently shown great promise for generative modeling, outperforming GANs on perceptual quality and autoregressive models at density estimation. A remaining downside is their slow sampling time: generating high quality samples takes many hundreds or thousands of model evaluations. Here we make two contributions to help eliminate this downside: First, we present new parameterizations of diffusion models that provide increased stability when using few sampling steps. Second, we present a method to distill a trained deterministic diffusion sampler, using many steps, into a new diffusion model that takes half as many sampling steps. We then keep progressively applying this distillation procedure to our model, halving the number of required sampling steps each time. On standard image generation benchmarks like CIFAR-10, ImageNet, and LSUN, we start out with state-of-the-art samplers taking as many as 8192 steps, and are able to distill down to models taking as few as 4 steps without losing much perceptual quality; achieving, for example, a FID of 3.0 on CIFAR-10 in 4 steps. Finally, we show that the full progressive distillation procedure does not take more time than it takes to train the original model, thus representing an efficient solution for generative modeling using diffusion at both train and test time.

Denoising diffusion (score-based) generative models have recently achieved significant accomplishments in generating realistic and diverse data. These approaches define a forward diffusion process for transforming data into noise and a backward denoising process for sampling data from noise. Unfortunately, the generation process of current denoising diffusion models is notoriously slow due to the lengthy iterative noise estimations, which rely on cumbersome neural networks. It prevents the diffusion models from being widely deployed, especially on edge devices. Previous works accelerate the generation process of diffusion model (DM) via finding shorter yet effective sampling trajectories. However, they overlook the cost of noise estimation with a heavy network in every iteration. In this work, we accelerate generation from the perspective of compressing the noise estimation network. Due to the difficulty of retraining DMs, we exclude mainstream training-aware compression paradigms and introduce post-training quantization (PTQ) into DM acceleration. However, the output distributions of noise estimation networks change with time-step, making previous PTQ methods fail in DMs since they are designed for single-time step scenarios. To devise a DM-specific PTQ method, we explore PTQ on DM in three aspects: quantized operations, calibration dataset, and calibration metric. We summarize and use several observations derived from all-inclusive investigations to formulate our method, which especially targets the unique multi-time-step structure of DMs. Experimentally, our method can directly quantize full-precision DMs into 8-bit models while maintaining or even improving their performance in a training-free manner. Importantly, our method can serve as a plug-and-play module on other fast-sampling methods, e.g., DDIM. The code is available at https://github.com/42Shawn/PTQ4DM .