Daily Papers

Recent work in Video Frame Interpolation (VFI) tries to formulate VFI as a diffusion-based conditional image generation problem, synthesizing the intermediate frame given a random noise and neighboring frames. Due to the relatively high resolution of videos, Latent Diffusion Models (LDMs) are employed as the conditional generation model, where the autoencoder compresses images into latent representations for diffusion and then reconstructs images from these latent representations. Such a formulation poses a crucial challenge: VFI expects that the output is deterministically equal to the ground truth intermediate frame, but LDMs randomly generate a diverse set of different images when the model runs multiple times. The reason for the diverse generation is that the cumulative variance (variance accumulated at each step of generation) of generated latent representations in LDMs is large. This makes the sampling trajectory random, resulting in diverse rather than deterministic generations. To address this problem, we propose our unique solution: Frame Interpolation with Consecutive Brownian Bridge Diffusion. Specifically, we propose consecutive Brownian Bridge diffusion that takes a deterministic initial value as input, resulting in a much smaller cumulative variance of generated latent representations. Our experiments suggest that our method can improve together with the improvement of the autoencoder and achieve state-of-the-art performance in VFI, leaving strong potential for further enhancement.

Video Frame Interpolation (VFI) aims to predict the intermediate frame I_n (we use n to denote time in videos to avoid notation overload with the timestep t in diffusion models) based on two consecutive neighboring frames I_0 and I_1. Recent approaches apply diffusion models (both image-based and video-based) in this task and achieve strong performance. However, image-based diffusion models are unable to extract temporal information and are relatively inefficient compared to non-diffusion methods. Video-based diffusion models can extract temporal information, but they are too large in terms of training scale, model size, and inference time. To mitigate the above issues, we propose Temporal-Aware Latent Brownian Bridge Diffusion for Video Frame Interpolation (TLB-VFI), an efficient video-based diffusion model. By extracting rich temporal information from video inputs through our proposed 3D-wavelet gating and temporal-aware autoencoder, our method achieves 20% improvement in FID on the most challenging datasets over recent SOTA of image-based diffusion models. Meanwhile, due to the existence of rich temporal information, our method achieves strong performance while having 3times fewer parameters. Such a parameter reduction results in 2.3x speed up. By incorporating optical flow guidance, our method requires 9000x less training data and achieves over 20x fewer parameters than video-based diffusion models. Codes and results are available at our project page: https://zonglinl.github.io/tlbvfi_page.

Denoising diffusion bridge models (DDBMs) are a powerful variant of diffusion models for interpolating between two arbitrary paired distributions given as endpoints. Despite their promising performance in tasks like image translation, DDBMs require a computationally intensive sampling process that involves the simulation of a (stochastic) differential equation through hundreds of network evaluations. In this work, we take the first step in fast sampling of DDBMs without extra training, motivated by the well-established recipes in diffusion models. We generalize DDBMs via a class of non-Markovian diffusion bridges defined on the discretized timesteps concerning sampling, which share the same marginal distributions and training objectives, give rise to generative processes ranging from stochastic to deterministic, and result in diffusion bridge implicit models (DBIMs). DBIMs are not only up to 25times faster than the vanilla sampler of DDBMs but also induce a novel, simple, and insightful form of ordinary differential equation (ODE) which inspires high-order numerical solvers. Moreover, DBIMs maintain the generation diversity in a distinguished way, by using a booting noise in the initial sampling step, which enables faithful encoding, reconstruction, and semantic interpolation in image translation tasks. Code is available at https://github.com/thu-ml/DiffusionBridge.

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.

Diffusion models are powerful generative models that map noise to data using stochastic processes. However, for many applications such as image editing, the model input comes from a distribution that is not random noise. As such, diffusion models must rely on cumbersome methods like guidance or projected sampling to incorporate this information in the generative process. In our work, we propose Denoising Diffusion Bridge Models (DDBMs), a natural alternative to this paradigm based on diffusion bridges, a family of processes that interpolate between two paired distributions given as endpoints. Our method learns the score of the diffusion bridge from data and maps from one endpoint distribution to the other by solving a (stochastic) differential equation based on the learned score. Our method naturally unifies several classes of generative models, such as score-based diffusion models and OT-Flow-Matching, allowing us to adapt existing design and architectural choices to our more general problem. Empirically, we apply DDBMs to challenging image datasets in both pixel and latent space. On standard image translation problems, DDBMs achieve significant improvement over baseline methods, and, when we reduce the problem to image generation by setting the source distribution to random noise, DDBMs achieve comparable FID scores to state-of-the-art methods despite being built for a more general task.

Diffusion bridges have shown potential in paired image-to-image (I2I) translation tasks. However, existing methods are limited by their unidirectional nature, requiring separate models for forward and reverse translations. This not only doubles the computational cost but also restricts their practicality. In this work, we introduce the Bidirectional Diffusion Bridge Model (BDBM), a scalable approach that facilitates bidirectional translation between two coupled distributions using a single network. BDBM leverages the Chapman-Kolmogorov Equation for bridges, enabling it to model data distribution shifts across timesteps in both forward and backward directions by exploiting the interchangeability of the initial and target timesteps within this framework. Notably, when the marginal distribution given endpoints is Gaussian, BDBM's transition kernels in both directions possess analytical forms, allowing for efficient learning with a single network. We demonstrate the connection between BDBM and existing bridge methods, such as Doob's h-transform and variational approaches, and highlight its advantages. Extensive experiments on high-resolution I2I translation tasks demonstrate that BDBM not only enables bidirectional translation with minimal additional cost but also outperforms state-of-the-art bridge models. Our source code is available at [https://github.com/kvmduc/BDBM||https://github.com/kvmduc/BDBM].

Diffusion model-based inverse problem solvers have shown impressive performance, but are limited in speed, mostly as they require reverse diffusion sampling starting from noise. Several recent works have tried to alleviate this problem by building a diffusion process, directly bridging the clean and the corrupted for specific inverse problems. In this paper, we first unify these existing works under the name Direct Diffusion Bridges (DDB), showing that while motivated by different theories, the resulting algorithms only differ in the choice of parameters. Then, we highlight a critical limitation of the current DDB framework, namely that it does not ensure data consistency. To address this problem, we propose a modified inference procedure that imposes data consistency without the need for fine-tuning. We term the resulting method data Consistent DDB (CDDB), which outperforms its inconsistent counterpart in terms of both perception and distortion metrics, thereby effectively pushing the Pareto-frontier toward the optimum. Our proposed method achieves state-of-the-art results on both evaluation criteria, showcasing its superiority over existing methods.

Diffusion models are a powerful class of generative models which simulate stochastic differential equations (SDEs) to generate data from noise. While diffusion models have achieved remarkable progress, they have limitations in unpaired image-to-image (I2I) translation tasks due to the Gaussian prior assumption. Schr\"{o}dinger Bridge (SB), which learns an SDE to translate between two arbitrary distributions, have risen as an attractive solution to this problem. Yet, to our best knowledge, none of SB models so far have been successful at unpaired translation between high-resolution images. In this work, we propose Unpaired Neural Schr\"{o}dinger Bridge (UNSB), which expresses the SB problem as a sequence of adversarial learning problems. This allows us to incorporate advanced discriminators and regularization to learn a SB between unpaired data. We show that UNSB is scalable and successfully solves various unpaired I2I translation tasks. Code: https://github.com/cyclomon/UNSB

Diffusion models have recently achieved success in solving Bayesian inverse problems with learned data priors. Current methods build on top of the diffusion sampling process, where each denoising step makes small modifications to samples from the previous step. However, this process struggles to correct errors from earlier sampling steps, leading to worse performance in complicated nonlinear inverse problems, such as phase retrieval. To address this challenge, we propose a new method called Decoupled Annealing Posterior Sampling (DAPS) that relies on a novel noise annealing process. Specifically, we decouple consecutive steps in a diffusion sampling trajectory, allowing them to vary considerably from one another while ensuring their time-marginals anneal to the true posterior as we reduce noise levels. This approach enables the exploration of a larger solution space, improving the success rate for accurate reconstructions. We demonstrate that DAPS significantly improves sample quality and stability across multiple image restoration tasks, particularly in complicated nonlinear inverse problems. For example, we achieve a PSNR of 30.72dB on the FFHQ 256 dataset for phase retrieval, which is an improvement of 9.12dB compared to existing methods.

We introduce a theoretical framework for diffusion-based image editing by formulating it as a reverse-time bridge modeling problem. This approach modifies the backward process of a pretrained diffusion model to construct a bridge that converges to an implicit distribution associated with the editing target at time 0. Building on this framework, we propose h-Edit, a novel editing method that utilizes Doob's h-transform and Langevin Monte Carlo to decompose the update of an intermediate edited sample into two components: a "reconstruction" term and an "editing" term. This decomposition provides flexibility, allowing the reconstruction term to be computed via existing inversion techniques and enabling the combination of multiple editing terms to handle complex editing tasks. To our knowledge, h-Edit is the first training-free method capable of performing simultaneous text-guided and reward-model-based editing. Extensive experiments, both quantitative and qualitative, show that h-Edit outperforms state-of-the-art baselines in terms of editing effectiveness and faithfulness. Our source code is available at https://github.com/nktoan/h-edit.

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.

The Schr\"odinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-time stochastic processes. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations. To address this limitation, we propose a novel Discrete-time IMF (D-IMF) procedure in which learning of stochastic processes is replaced by learning just a few transition probabilities in discrete time. Its great advantage is that in practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), an already well-established adversarial generative modeling technique. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds. We provide the code at https://github.com/Daniil-Selikhanovych/ASBM.

Designing biological sequences is an important challenge that requires satisfying complex constraints and thus is a natural problem to address with deep generative modeling. Diffusion generative models have achieved considerable success in many applications. Score-based generative stochastic differential equations (SDE) model is a continuous-time diffusion model framework that enjoys many benefits, but the originally proposed SDEs are not naturally designed for modeling discrete data. To develop generative SDE models for discrete data such as biological sequences, here we introduce a diffusion process defined in the probability simplex space with stationary distribution being the Dirichlet distribution. This makes diffusion in continuous space natural for modeling discrete data. We refer to this approach as Dirchlet diffusion score model. We demonstrate that this technique can generate samples that satisfy hard constraints using a Sudoku generation task. This generative model can also solve Sudoku, including hard puzzles, without additional training. Finally, we applied this approach to develop the first human promoter DNA sequence design model and showed that designed sequences share similar properties with natural promoter sequences.

Learning diffusion bridge models is easy; making them fast and practical is an art. Diffusion bridge models (DBMs) are a promising extension of diffusion models for applications in image-to-image translation. However, like many modern diffusion and flow models, DBMs suffer from the problem of slow inference. To address it, we propose a novel distillation technique based on the inverse bridge matching formulation and derive the tractable objective to solve it in practice. Unlike previously developed DBM distillation techniques, the proposed method can distill both conditional and unconditional types of DBMs, distill models in a one-step generator, and use only the corrupted images for training. We evaluate our approach for both conditional and unconditional types of bridge matching on a wide set of setups, including super-resolution, JPEG restoration, sketch-to-image, and other tasks, and show that our distillation technique allows us to accelerate the inference of DBMs from 4x to 100x and even provide better generation quality than used teacher model depending on particular setup.

Common image-to-image translation methods rely on joint training over data from both source and target domains. The training process requires concurrent access to both datasets, which hinders data separation and privacy protection; and existing models cannot be easily adapted for translation of new domain pairs. We present Dual Diffusion Implicit Bridges (DDIBs), an image translation method based on diffusion models, that circumvents training on domain pairs. Image translation with DDIBs relies on two diffusion models trained independently on each domain, and is a two-step process: DDIBs first obtain latent encodings for source images with the source diffusion model, and then decode such encodings using the target model to construct target images. Both steps are defined via ordinary differential equations (ODEs), thus the process is cycle consistent only up to discretization errors of the ODE solvers. Theoretically, we interpret DDIBs as concatenation of source to latent, and latent to target Schrodinger Bridges, a form of entropy-regularized optimal transport, to explain the efficacy of the method. Experimentally, we apply DDIBs on synthetic and high-resolution image datasets, to demonstrate their utility in a wide variety of translation tasks and their inherent optimal transport properties.

Diffusion models have become the go-to method for large-scale generative models in real-world applications. These applications often involve data distributions confined within bounded domains, typically requiring ad-hoc thresholding techniques for boundary enforcement. Reflected diffusion models (Lou23) aim to enhance generalizability by generating the data distribution through a backward process governed by reflected Brownian motion. However, reflected diffusion models may not easily adapt to diverse domains without the derivation of proper diffeomorphic mappings and do not guarantee optimal transport properties. To overcome these limitations, we introduce the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive elegant reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and explore natural connections to entropic optimal transport for the study of approximate linear convergence - a valuable insight for practical training. Our algorithm yields robust generative modeling in diverse domains, and its scalability is demonstrated in real-world constrained generative modeling through standard image benchmarks.

Retrosynthesis planning is a fundamental challenge in chemistry which aims at designing reaction pathways from commercially available starting materials to a target molecule. Each step in multi-step retrosynthesis planning requires accurate prediction of possible precursor molecules given the target molecule and confidence estimates to guide heuristic search algorithms. We model single-step retrosynthesis planning as a distribution learning problem in a discrete state space. First, we introduce the Markov Bridge Model, a generative framework aimed to approximate the dependency between two intractable discrete distributions accessible via a finite sample of coupled data points. Our framework is based on the concept of a Markov bridge, a Markov process pinned at its endpoints. Unlike diffusion-based methods, our Markov Bridge Model does not need a tractable noise distribution as a sampling proxy and directly operates on the input product molecules as samples from the intractable prior distribution. We then address the retrosynthesis planning problem with our novel framework and introduce RetroBridge, a template-free retrosynthesis modeling approach that achieves state-of-the-art results on standard evaluation benchmarks.

Connecting optimal transport and variational inference, we present a principled and systematic framework for sampling and generative modelling centred around divergences on path space. Our work culminates in the development of the Controlled Monte Carlo Diffusion sampler (CMCD) for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. On the way, we clarify the relationship between the EM-algorithm and iterative proportional fitting (IPF) for Schr{\"o}dinger bridges, deriving as well a regularised objective that bypasses the iterative bottleneck of standard IPF-updates. Finally, we show that CMCD has a strong foundation in the Jarzinsky and Crooks identities from statistical physics, and that it convincingly outperforms competing approaches across a wide array of experiments.

In this work, we consider rather general and broad class of Markov chains, Ito chains, that look like Euler-Maryama discretization of some Stochastic Differential Equation. The chain we study is a unified framework for theoretical analysis. It comes with almost arbitrary isotropic and state-dependent noise instead of normal and state-independent one as in most related papers. Moreover, in our chain the drift and diffusion coefficient can be inexact in order to cover wide range of applications as Stochastic Gradient Langevin Dynamics, sampling, Stochastic Gradient Descent or Stochastic Gradient Boosting. We prove the bound in W_{2}-distance between the laws of our Ito chain and corresponding differential equation. These results improve or cover most of the known estimates. And for some particular cases, our analysis is the first.

Recent studies have demonstrated the strong empirical performance of diffusion models on discrete sequences across domains from natural language to biological sequence generation. For example, in the protein inverse folding task, conditional diffusion models have achieved impressive results in generating natural-like sequences that fold back into the original structure. However, practical design tasks often require not only modeling a conditional distribution but also optimizing specific task objectives. For instance, we may prefer protein sequences with high stability. To address this, we consider the scenario where we have pre-trained discrete diffusion models that can generate natural-like sequences, as well as reward models that map sequences to task objectives. We then formulate the reward maximization problem within discrete diffusion models, analogous to reinforcement learning (RL), while minimizing the KL divergence against pretrained diffusion models to preserve naturalness. To solve this RL problem, we propose a novel algorithm, DRAKES, that enables direct backpropagation of rewards through entire trajectories generated by diffusion models, by making the originally non-differentiable trajectories differentiable using the Gumbel-Softmax trick. Our theoretical analysis indicates that our approach can generate sequences that are both natural-like and yield high rewards. While similar tasks have been recently explored in diffusion models for continuous domains, our work addresses unique algorithmic and theoretical challenges specific to discrete diffusion models, which arise from their foundation in continuous-time Markov chains rather than Brownian motion. Finally, we demonstrate the effectiveness of DRAKES in generating DNA and protein sequences that optimize enhancer activity and protein stability, respectively, important tasks for gene therapies and protein-based therapeutics.

Diffusion (score-based) generative models have been widely used for modeling various types of complex data, including images, audios, and point clouds. Recently, the deep connection between forward-backward stochastic differential equations (SDEs) and diffusion-based models has been revealed, and several new variants of SDEs are proposed (e.g., sub-VP, critically-damped Langevin) along this line. Despite the empirical success of the hand-crafted fixed forward SDEs, a great quantity of proper forward SDEs remain unexplored. In this work, we propose a general framework for parameterizing the diffusion model, especially the spatial part of the forward SDE. An abstract formalism is introduced with theoretical guarantees, and its connection with previous diffusion models is leveraged. We demonstrate the theoretical advantage of our method from an optimization perspective. Numerical experiments on synthetic datasets, MINIST and CIFAR10 are also presented to validate the effectiveness of our framework.

Conventional diffusion models typically relies on a fixed forward process, which implicitly defines complex marginal distributions over latent variables. This can often complicate the reverse process' task in learning generative trajectories, and results in costly inference for diffusion models. To address these limitations, we introduce Neural Flow Diffusion Models (NFDM), a novel framework that enhances diffusion models by supporting a broader range of forward processes beyond the fixed linear Gaussian. We also propose a novel parameterization technique for learning the forward process. Our framework provides an end-to-end, simulation-free optimization objective, effectively minimizing a variational upper bound on the negative log-likelihood. Experimental results demonstrate NFDM's strong performance, evidenced by state-of-the-art likelihood estimation. Furthermore, we investigate NFDM's capacity for learning generative dynamics with specific characteristics, such as deterministic straight lines trajectories. This exploration underscores NFDM's versatility and its potential for a wide range of applications.

Generating graph-structured data requires learning the underlying distribution of graphs. Yet, this is a challenging problem, and the previous graph generative methods either fail to capture the permutation-invariance property of graphs or cannot sufficiently model the complex dependency between nodes and edges, which is crucial for generating real-world graphs such as molecules. To overcome such limitations, we propose a novel score-based generative model for graphs with a continuous-time framework. Specifically, we propose a new graph diffusion process that models the joint distribution of the nodes and edges through a system of stochastic differential equations (SDEs). Then, we derive novel score matching objectives tailored for the proposed diffusion process to estimate the gradient of the joint log-density with respect to each component, and introduce a new solver for the system of SDEs to efficiently sample from the reverse diffusion process. We validate our graph generation method on diverse datasets, on which it either achieves significantly superior or competitive performance to the baselines. Further analysis shows that our method is able to generate molecules that lie close to the training distribution yet do not violate the chemical valency rule, demonstrating the effectiveness of the system of SDEs in modeling the node-edge relationships. Our code is available at https://github.com/harryjo97/GDSS.

Diffusion models for continuous data gained widespread adoption owing to their high quality generation and control mechanisms. However, controllable diffusion on discrete data faces challenges given that continuous guidance methods do not directly apply to discrete diffusion. Here, we provide a straightforward derivation of classifier-free and classifier-based guidance for discrete diffusion, as well as a new class of diffusion models that leverage uniform noise and that are more guidable because they can continuously edit their outputs. We improve the quality of these models with a novel continuous-time variational lower bound that yields state-of-the-art performance, especially in settings involving guidance or fast generation. Empirically, we demonstrate that our guidance mechanisms combined with uniform noise diffusion improve controllable generation relative to autoregressive and diffusion baselines on several discrete data domains, including genomic sequences, small molecule design, and discretized image generation.

Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

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.

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, 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.

Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on manifolds suffer from expensive divergence computation or rely on approximations of heat kernel. These limitations restrict their applicability to simple geometries and hinder scalability to high dimensions. In this work, we introduce the Riemannian Diffusion Mixture, a principled framework for building a generative diffusion process on manifolds. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes derived on general manifolds without requiring heat kernel estimations. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points that guides the process toward the data distribution. We further propose a scalable training objective for learning the mixture process that readily applies to general manifolds. Our method achieves superior performance on diverse manifolds with dramatically reduced number of in-training simulation steps for general manifolds.

Discrete diffusion has recently emerged as a promising paradigm in discrete data modeling. However, existing methods typically rely on a fixed rate transition matrix during training, which not only limits the expressiveness of latent representations, a fundamental strength of variational methods, but also constrains the overall design space. To address these limitations, we propose Discrete Markov Bridge, a novel framework specifically designed for discrete representation learning. Our approach is built upon two key components: Matrix Learning and Score Learning. We conduct a rigorous theoretical analysis, establishing formal performance guarantees for Matrix Learning and proving the convergence of the overall framework. Furthermore, we analyze the space complexity of our method, addressing practical constraints identified in prior studies. Extensive empirical evaluations validate the effectiveness of the proposed Discrete Markov Bridge, which achieves an Evidence Lower Bound (ELBO) of 1.38 on the Text8 dataset, outperforming established baselines. Moreover, the proposed model demonstrates competitive performance on the CIFAR-10 dataset, achieving results comparable to those obtained by image-specific generation approaches.

In a convergence of machine learning and biology, we reveal that diffusion models are evolutionary algorithms. By considering evolution as a denoising process and reversed evolution as diffusion, we mathematically demonstrate that diffusion models inherently perform evolutionary algorithms, naturally encompassing selection, mutation, and reproductive isolation. Building on this equivalence, we propose the Diffusion Evolution method: an evolutionary algorithm utilizing iterative denoising -- as originally introduced in the context of diffusion models -- to heuristically refine solutions in parameter spaces. Unlike traditional approaches, Diffusion Evolution efficiently identifies multiple optimal solutions and outperforms prominent mainstream evolutionary algorithms. Furthermore, leveraging advanced concepts from diffusion models, namely latent space diffusion and accelerated sampling, we introduce Latent Space Diffusion Evolution, which finds solutions for evolutionary tasks in high-dimensional complex parameter space while significantly reducing computational steps. This parallel between diffusion and evolution not only bridges two different fields but also opens new avenues for mutual enhancement, raising questions about open-ended evolution and potentially utilizing non-Gaussian or discrete diffusion models in the context of Diffusion Evolution.

Although diffusion models (DMs) have shown promising performances in a number of tasks (e.g., speech synthesis and image generation), they might suffer from error propagation because of their sequential structure. However, this is not certain because some sequential models, such as Conditional Random Field (CRF), are free from this problem. To address this issue, we develop a theoretical framework to mathematically formulate error propagation in the architecture of DMs, The framework contains three elements, including modular error, cumulative error, and propagation equation. The modular and cumulative errors are related by the equation, which interprets that DMs are indeed affected by error propagation. Our theoretical study also suggests that the cumulative error is closely related to the generation quality of DMs. Based on this finding, we apply the cumulative error as a regularization term to reduce error propagation. Because the term is computationally intractable, we derive its upper bound and design a bootstrap algorithm to efficiently estimate the bound for optimization. We have conducted extensive experiments on multiple image datasets, showing that our proposed regularization reduces error propagation, significantly improves vanilla DMs, and outperforms previous baselines.

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.

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.

This paper introduces a novel theoretical simplification of the Diffusion Schr\"odinger Bridge (DSB) that facilitates its unification with Score-based Generative Models (SGMs), addressing the limitations of DSB in complex data generation and enabling faster convergence and enhanced performance. By employing SGMs as an initial solution for DSB, our approach capitalizes on the strengths of both frameworks, ensuring a more efficient training process and improving the performance of SGM. We also propose a reparameterization technique that, despite theoretical approximations, practically improves the network's fitting capabilities. Our extensive experimental evaluations confirm the effectiveness of the simplified DSB, demonstrating its significant improvements. We believe the contributions of this work pave the way for advanced generative modeling. The code is available at https://github.com/checkcrab/SDSB.

Being the most cutting-edge generative methods, diffusion methods have shown great advances in wide generation tasks. Among them, graph generation attracts significant research attention for its broad application in real life. In our survey, we systematically and comprehensively review on diffusion-based graph generative methods. We first make a review on three mainstream paradigms of diffusion methods, which are denoising diffusion probabilistic models, score-based genrative models, and stochastic differential equations. Then we further categorize and introduce the latest applications of diffusion models on graphs. In the end, we point out some limitations of current studies and future directions of future explorations. The summary of existing methods metioned in this survey is in https://github.com/zhejiangzhuque/Diffusion-based-Graph-Generative-Methods.

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.

Diffusion models (DMs) have been adopted across diverse fields with its remarkable abilities in capturing intricate data distributions. In this paper, we propose a Fast Diffusion Model (FDM) to significantly speed up DMs from a stochastic optimization perspective for both faster training and sampling. We first find that the diffusion process of DMs accords with the stochastic optimization process of stochastic gradient descent (SGD) on a stochastic time-variant problem. Then, inspired by momentum SGD that uses both gradient and an extra momentum to achieve faster and more stable convergence than SGD, we integrate momentum into the diffusion process of DMs. This comes with a unique challenge of deriving the noise perturbation kernel from the momentum-based diffusion process. To this end, we frame the process as a Damped Oscillation system whose critically damped state -- the kernel solution -- avoids oscillation and yields a faster convergence speed of the diffusion process. Empirical results show that our FDM can be applied to several popular DM frameworks, e.g., VP, VE, and EDM, and reduces their training cost by about 50% with comparable image synthesis performance on CIFAR-10, FFHQ, and AFHQv2 datasets. Moreover, FDM decreases their sampling steps by about 3x to achieve similar performance under the same samplers. The code is available at https://github.com/sail-sg/FDM.

Diffusion model-based inverse problem solvers have demonstrated state-of-the-art performance in cases where the forward operator is known (i.e. non-blind). However, the applicability of the method to blind inverse problems has yet to be explored. In this work, we show that we can indeed solve a family of blind inverse problems by constructing another diffusion prior for the forward operator. Specifically, parallel reverse diffusion guided by gradients from the intermediate stages enables joint optimization of both the forward operator parameters as well as the image, such that both are jointly estimated at the end of the parallel reverse diffusion procedure. We show the efficacy of our method on two representative tasks -- blind deblurring, and imaging through turbulence -- and show that our method yields state-of-the-art performance, while also being flexible to be applicable to general blind inverse problems when we know the functional forms.

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative function distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression to determine optimal approximation coefficients. Furthermore, we propose the use of neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,-an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.

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.

The present work investigates two properties of level crossings of a stationary Gaussian process X(t) with autocorrelation function R_X(tau). We show firstly that if R_X(tau) admits finite second and fourth derivatives at the origin, the length of up-excursions above a large negative level -gamma is asymptotically exponential as -gamma to -infty. Secondly, assuming that R_X(tau) admits a finite second derivative at the origin and some defined properties, we derive the mean number of crossings as well as the length of successive excursions above two subsequent large levels. The asymptotic results are shown to be effective even for moderate values of crossing level. An application of the developed results is proposed to derive the probability of successive excursions above adjacent levels during a time window.

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) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in phase space dynamics, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

Denoising diffusion models, a class of generative models, have garnered immense interest lately in various deep-learning problems. A diffusion probabilistic model defines a forward diffusion stage where the input data is gradually perturbed over several steps by adding Gaussian noise and then learns to reverse the diffusion process to retrieve the desired noise-free data from noisy data samples. Diffusion models are widely appreciated for their strong mode coverage and quality of the generated samples despite their known computational burdens. Capitalizing on the advances in computer vision, the field of medical imaging has also observed a growing interest in diffusion models. To help the researcher navigate this profusion, this survey intends to provide a comprehensive overview of diffusion models in the discipline of medical image analysis. Specifically, we introduce the solid theoretical foundation and fundamental concepts behind diffusion models and the three generic diffusion modelling frameworks: diffusion probabilistic models, noise-conditioned score networks, and stochastic differential equations. Then, we provide a systematic taxonomy of diffusion models in the medical domain and propose a multi-perspective categorization based on their application, imaging modality, organ of interest, and algorithms. To this end, we cover extensive applications of diffusion models in the medical domain. Furthermore, we emphasize the practical use case of some selected approaches, and then we discuss the limitations of the diffusion models in the medical domain and propose several directions to fulfill the demands of this field. Finally, we gather the overviewed studies with their available open-source implementations at https://github.com/amirhossein-kz/Awesome-Diffusion-Models-in-Medical-Imaging.

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 gradual nature of a diffusion process that synthesizes samples in small increments constitutes a key ingredient of Denoising Diffusion Probabilistic Models (DDPM), which have presented unprecedented quality in image synthesis and been recently explored in the motion domain. In this work, we propose to adapt the gradual diffusion concept (operating along a diffusion time-axis) into the temporal-axis of the motion sequence. Our key idea is to extend the DDPM framework to support temporally varying denoising, thereby entangling the two axes. Using our special formulation, we iteratively denoise a motion buffer that contains a set of increasingly-noised poses, which auto-regressively produces an arbitrarily long stream of frames. With a stationary diffusion time-axis, in each diffusion step we increment only the temporal-axis of the motion such that the framework produces a new, clean frame which is removed from the beginning of the buffer, followed by a newly drawn noise vector that is appended to it. This new mechanism paves the way towards a new framework for long-term motion synthesis with applications to character animation and other domains.

We derive a minimalist but powerful deterministic denoising-diffusion model. While denoising diffusion has shown great success in many domains, its underlying theory remains largely inaccessible to non-expert users. Indeed, an understanding of graduate-level concepts such as Langevin dynamics or score matching appears to be required to grasp how it works. We propose an alternative approach that requires no more than undergrad calculus and probability. We consider two densities and observe what happens when random samples from these densities are blended (linearly interpolated). We show that iteratively blending and deblending samples produces random paths between the two densities that converge toward a deterministic mapping. This mapping can be evaluated with a neural network trained to deblend samples. We obtain a model that behaves like deterministic denoising diffusion: it iteratively maps samples from one density (e.g., Gaussian noise) to another (e.g., cat images). However, compared to the state-of-the-art alternative, our model is simpler to derive, simpler to implement, more numerically stable, achieves higher quality results in our experiments, and has interesting connections to computer graphics.

In text-to-speech (TTS) synthesis, diffusion models have achieved promising generation quality. However, because of the pre-defined data-to-noise diffusion process, their prior distribution is restricted to a noisy representation, which provides little information of the generation target. In this work, we present a novel TTS system, Bridge-TTS, making the first attempt to substitute the noisy Gaussian prior in established diffusion-based TTS methods with a clean and deterministic one, which provides strong structural information of the target. Specifically, we leverage the latent representation obtained from text input as our prior, and build a fully tractable Schrodinger bridge between it and the ground-truth mel-spectrogram, leading to a data-to-data process. Moreover, the tractability and flexibility of our formulation allow us to empirically study the design spaces such as noise schedules, as well as to develop stochastic and deterministic samplers. Experimental results on the LJ-Speech dataset illustrate the effectiveness of our method in terms of both synthesis quality and sampling efficiency, significantly outperforming our diffusion counterpart Grad-TTS in 50-step/1000-step synthesis and strong fast TTS models in few-step scenarios. Project page: https://bridge-tts.github.io/

Diffusion models have quickly risen in popularity for their ability to model complex distributions and perform effective posterior sampling. Unfortunately, the iterative nature of these generative models makes them computationally expensive and unsuitable for real-time sequential inverse problems such as ultrasound imaging. Considering the strong temporal structure across sequences of frames, we propose a novel approach that models the transition dynamics to improve the efficiency of sequential diffusion posterior sampling in conditional image synthesis. Through modeling sequence data using a video vision transformer (ViViT) transition model based on previous diffusion outputs, we can initialize the reverse diffusion trajectory at a lower noise scale, greatly reducing the number of iterations required for convergence. We demonstrate the effectiveness of our approach on a real-world dataset of high frame rate cardiac ultrasound images and show that it achieves the same performance as a full diffusion trajectory while accelerating inference 25times, enabling real-time posterior sampling. Furthermore, we show that the addition of a transition model improves the PSNR up to 8\% in cases with severe motion. Our method opens up new possibilities for real-time applications of diffusion models in imaging and other domains requiring real-time inference.

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-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.

Diffusion models have demonstrated remarkable success in generative tasks, yet their iterative denoising process results in slow inference, limiting their practicality. While existing acceleration methods exploit the well-known U-shaped similarity pattern between adjacent steps through caching mechanisms, they lack theoretical foundation and rely on simplistic computation reuse, often leading to performance degradation. In this work, we provide a theoretical understanding by analyzing the denoising process through the second-order Adams-Bashforth method, revealing a linear relationship between the outputs of consecutive steps. This analysis explains why the outputs of adjacent steps exhibit a U-shaped pattern. Furthermore, extending Adams-Bashforth method to higher order, we propose a novel caching-based acceleration approach for diffusion models, instead of directly reusing cached results, with a truncation error bound of only \(O(h^k)\) where h is the step size. Extensive validation across diverse image and video diffusion models (including HunyuanVideo and FLUX.1-dev) with various schedulers demonstrates our method's effectiveness in achieving nearly 3times speedup while maintaining original performance levels, offering a practical real-time solution without compromising generation quality.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

Diffusion models have recently been increasingly applied to temporal data such as video, fluid mechanics simulations, or climate data. These methods generally treat subsequent frames equally regarding the amount of noise in the diffusion process. This paper explores Rolling Diffusion: a new approach that uses a sliding window denoising process. It ensures that the diffusion process progressively corrupts through time by assigning more noise to frames that appear later in a sequence, reflecting greater uncertainty about the future as the generation process unfolds. Empirically, we show that when the temporal dynamics are complex, Rolling Diffusion is superior to standard diffusion. In particular, this result is demonstrated in a video prediction task using the Kinetics-600 video dataset and in a chaotic fluid dynamics forecasting experiment.

Diffusion models have emerged as a key pillar of foundation models in visual domains. One of their critical applications is to universally solve different downstream inverse tasks via a single diffusion prior without re-training for each task. Most inverse tasks can be formulated as inferring a posterior distribution over data (e.g., a full image) given a measurement (e.g., a masked image). This is however challenging in diffusion models since the nonlinear and iterative nature of the diffusion process renders the posterior intractable. To cope with this challenge, we propose a variational approach that by design seeks to approximate the true posterior distribution. We show that our approach naturally leads to regularization by denoising diffusion process (RED-Diff) where denoisers at different timesteps concurrently impose different structural constraints over the image. To gauge the contribution of denoisers from different timesteps, we propose a weighting mechanism based on signal-to-noise-ratio (SNR). Our approach provides a new variational perspective for solving inverse problems with diffusion models, allowing us to formulate sampling as stochastic optimization, where one can simply apply off-the-shelf solvers with lightweight iterates. Our experiments for image restoration tasks such as inpainting and superresolution demonstrate the strengths of our method compared with state-of-the-art sampling-based diffusion models.

We introduce Interleaved Gibbs Diffusion (IGD), a novel generative modeling framework for mixed continuous-discrete data, focusing on constrained generation problems. Prior works on discrete and continuous-discrete diffusion models assume factorized denoising distribution for fast generation, which can hinder the modeling of strong dependencies between random variables encountered in constrained generation. IGD moves beyond this by interleaving continuous and discrete denoising algorithms via a discrete time Gibbs sampling type Markov chain. IGD provides flexibility in the choice of denoisers, allows conditional generation via state-space doubling and inference time scaling via the ReDeNoise method. Empirical evaluations on three challenging tasks-solving 3-SAT, generating molecule structures, and generating layouts-demonstrate state-of-the-art performance. Notably, IGD achieves a 7% improvement on 3-SAT out of the box and achieves state-of-the-art results in molecule generation without relying on equivariant diffusion or domain-specific architectures. We explore a wide range of modeling, and interleaving strategies along with hyperparameters in each of these problems.

Diffusion models have established themselves as state-of-the-art generative models across various data modalities, including images and videos, due to their ability to accurately approximate complex data distributions. Unlike traditional generative approaches such as VAEs and GANs, diffusion models employ a progressive denoising process that transforms noise into meaningful data over multiple iterative steps. This gradual approach enhances their expressiveness and generation quality. Not only that, diffusion models have also been shown to extract meaningful representations from data while learning to generate samples. Despite their success, the application of diffusion models to graph-structured data remains relatively unexplored, primarily due to the discrete nature of graphs, which necessitates discrete diffusion processes distinct from the continuous methods used in other domains. In this work, we leverage the representational capabilities of diffusion models to learn meaningful embeddings for graph data. By training a discrete diffusion model within an autoencoder framework, we enable both effective autoencoding and representation learning tailored to the unique characteristics of graph-structured data. We only need the encoder at the end to extract representations. Our approach demonstrates the potential of discrete diffusion models to be used for graph representation learning.

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 .

Practitioners frequently aim to infer an unobserved population trajectory using sample snapshots at multiple time points. For instance, in single-cell sequencing, scientists would like to learn how gene expression evolves over time. But sequencing any cell destroys that cell. So we cannot access any cell's full trajectory, but we can access snapshot samples from many cells. Stochastic differential equations are commonly used to analyze systems with full individual-trajectory access; since here we have only sample snapshots, these methods are inapplicable. The deep learning community has recently explored using Schr\"odinger bridges (SBs) and their extensions to estimate these dynamics. However, these methods either (1) interpolate between just two time points or (2) require a single fixed reference dynamic within the SB, which is often just set to be Brownian motion. But learning piecewise from adjacent time points can fail to capture long-term dependencies. And practitioners are typically able to specify a model class for the reference dynamic but not the exact values of the parameters within it. So we propose a new method that (1) learns the unobserved trajectories from sample snapshots across multiple time points and (2) requires specification only of a class of reference dynamics, not a single fixed one. In particular, we suggest an iterative projection method inspired by Schr\"odinger bridges; we alternate between learning a piecewise SB on the unobserved trajectories and using the learned SB to refine our best guess for the dynamics within the reference class. We demonstrate the advantages of our method via a well-known simulated parametric model from ecology, simulated and real data from systems biology, and real motion-capture data.

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).

Diffusion models have emerged as a promising alternative to autoregressive models in modeling discrete categorical data. Yet diffusion models that directly work on discrete data space do not fully exploit the power of iterative refinement, as the signals are lost during the transition between discrete states. Existing continuous diffusion models for discrete data have limited performance compared to discrete approaches, and the unclear link between them restricts the development of diffusion models for discrete data. In this work, we propose a continuous diffusion model for language modeling that incorporates the geometry of the underlying categorical distribution. We establish a connection between the discrete diffusion and continuous flow on the statistical manifold, and building on the analogy, we introduce a simple design for the diffusion process that generalizes previous discrete diffusion models. We further propose a simulation-free training framework based on radial symmetry and a simple technique to address the high dimensionality of the manifold. Comprehensive experiments on language modeling benchmarks and other modalities show that our method outperforms existing discrete diffusion models and approaches the performance of autoregressive models. Codes available at https://github.com/harryjo97/RDLM{https://github.com/harryjo97/RDLM}.

In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation ability of idealised denoising diffusion models for high-dimensional target data, implementations introduce thresholding procedures for the generating process to overcome issues arising from the unbounded state space of such models. This mismatch between theoretical design and implementation of diffusion models has been addressed empirically by using a reflected diffusion process as the driver of noise instead. In this paper, we study statistical guarantees of these denoising reflected diffusion models. In particular, we establish minimax optimal rates of convergence in total variation, up to a polylogarithmic factor, under Sobolev smoothness assumptions. Our main contributions include the statistical analysis of this novel class of denoising reflected diffusion models and a refined score approximation method in both time and space, leveraging spectral decomposition and rigorous neural network analysis.

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

Uniform-state discrete diffusion models hold the promise of fast text generation due to their inherent ability to self-correct. However, they are typically outperformed by autoregressive models and masked diffusion models. In this work, we narrow this performance gap by leveraging a key insight: Uniform-state diffusion processes naturally emerge from an underlying Gaussian diffusion. Our method, Duo, transfers powerful techniques from Gaussian diffusion to improve both training and sampling. First, we introduce a curriculum learning strategy guided by the Gaussian process, doubling training speed by reducing variance. Models trained with curriculum learning surpass autoregressive models in zero-shot perplexity on 3 of 7 benchmarks. Second, we present Discrete Consistency Distillation, which adapts consistency distillation from the continuous to the discrete setting. This algorithm unlocks few-step generation in diffusion language models by accelerating sampling by two orders of magnitude. We provide the code and model checkpoints on the project page: http://s-sahoo.github.io/duo

Denoising diffusion probabilistic models (DDPMs) (Ho et al. 2020) have shown impressive results on image and waveform generation in continuous state spaces. Here, we introduce Discrete Denoising Diffusion Probabilistic Models (D3PMs), diffusion-like generative models for discrete data that generalize the multinomial diffusion model of Hoogeboom et al. 2021, by going beyond corruption processes with uniform transition probabilities. This includes corruption with transition matrices that mimic Gaussian kernels in continuous space, matrices based on nearest neighbors in embedding space, and matrices that introduce absorbing states. The third allows us to draw a connection between diffusion models and autoregressive and mask-based generative models. We show that the choice of transition matrix is an important design decision that leads to improved results in image and text domains. We also introduce a new loss function that combines the variational lower bound with an auxiliary cross entropy loss. For text, this model class achieves strong results on character-level text generation while scaling to large vocabularies on LM1B. On the image dataset CIFAR-10, our models approach the sample quality and exceed the log-likelihood of the continuous-space DDPM model.

This work introduces DiGress, a discrete denoising diffusion model for generating graphs with categorical node and edge attributes. Our model utilizes a discrete diffusion process that progressively edits graphs with noise, through the process of adding or removing edges and changing the categories. A graph transformer network is trained to revert this process, simplifying the problem of distribution learning over graphs into a sequence of node and edge classification tasks. We further improve sample quality by introducing a Markovian noise model that preserves the marginal distribution of node and edge types during diffusion, and by incorporating auxiliary graph-theoretic features. A procedure for conditioning the generation on graph-level features is also proposed. DiGress achieves state-of-the-art performance on molecular and non-molecular datasets, with up to 3x validity improvement on a planar graph dataset. It is also the first model to scale to the large GuacaMol dataset containing 1.3M drug-like molecules without the use of molecule-specific representations.

Recently, various methods have been proposed to address the inconsistency issue of DDIM inversion to enable image editing, such as EDICT [36] and Null-text inversion [22]. However, the above methods introduce considerable computational overhead. In this paper, we propose a new technique, named bi-directional integration approximation (BDIA), to perform exact diffusion inversion with neglible computational overhead. Suppose we would like to estimate the next diffusion state z_{i-1} at timestep t_i with the historical information (i,z_i) and (i+1,z_{i+1}). We first obtain the estimated Gaussian noise boldsymbol{epsilon}(z_i,i), and then apply the DDIM update procedure twice for approximating the ODE integration over the next time-slot [t_i, t_{i-1}] in the forward manner and the previous time-slot [t_i, t_{t+1}] in the backward manner. The DDIM step for the previous time-slot is used to refine the integration approximation made earlier when computing z_i. A nice property of BDIA-DDIM is that the update expression for z_{i-1} is a linear combination of (z_{i+1}, z_i, boldsymbol{epsilon}(z_i,i)). This allows for exact backward computation of z_{i+1} given (z_i, z_{i-1}), thus leading to exact diffusion inversion. It is demonstrated with experiments that (round-trip) BDIA-DDIM is particularly effective for image editing. Our experiments further show that BDIA-DDIM produces markedly better image sampling qualities than DDIM for text-to-image generation. BDIA can also be applied to improve the performance of other ODE solvers in addition to DDIM. In our work, it is found that applying BDIA to the EDM sampling procedure produces consistently better performance over four pre-trained models.

Generation of graphs is a major challenge for real-world tasks that require understanding the complex nature of their non-Euclidean structures. Although diffusion models have achieved notable success in graph generation recently, they are ill-suited for modeling the topological properties of graphs since learning to denoise the noisy samples does not explicitly learn the graph structures to be generated. To tackle this limitation, we propose a generative framework that models the topology of graphs by explicitly learning the final graph structures of the diffusion process. Specifically, we design the generative process as a mixture of endpoint-conditioned diffusion processes which is driven toward the predicted graph that results in rapid convergence. We further introduce a simple parameterization of the mixture process and develop an objective for learning the final graph structure, which enables maximum likelihood training. Through extensive experimental validation on general graph and 2D/3D molecule generation tasks, we show that our method outperforms previous generative models, generating graphs with correct topology with both continuous (e.g. 3D coordinates) and discrete (e.g. atom types) features. Our code is available at https://github.com/harryjo97/GruM.

Background: Deep learning models are typically trained using stochastic gradient descent or one of its variants. These methods update the weights using their gradient, estimated from a small fraction of the training data. It has been observed that when using large batch sizes there is a persistent degradation in generalization performance - known as the "generalization gap" phenomena. Identifying the origin of this gap and closing it had remained an open problem. Contributions: We examine the initial high learning rate training phase. We find that the weight distance from its initialization grows logarithmically with the number of weight updates. We therefore propose a "random walk on random landscape" statistical model which is known to exhibit similar "ultra-slow" diffusion behavior. Following this hypothesis we conducted experiments to show empirically that the "generalization gap" stems from the relatively small number of updates rather than the batch size, and can be completely eliminated by adapting the training regime used. We further investigate different techniques to train models in the large-batch regime and present a novel algorithm named "Ghost Batch Normalization" which enables significant decrease in the generalization gap without increasing the number of updates. To validate our findings we conduct several additional experiments on MNIST, CIFAR-10, CIFAR-100 and ImageNet. Finally, we reassess common practices and beliefs concerning training of deep models and suggest they may not be optimal to achieve good generalization.

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 have garnered significant interest from the community for their great generative ability across various applications. However, their typical multi-step sequential-denoising nature gives rise to high cumulative latency, thereby precluding the possibilities of parallel computation. To address this, we introduce AsyncDiff, a universal and plug-and-play acceleration scheme that enables model parallelism across multiple devices. Our approach divides the cumbersome noise prediction model into multiple components, assigning each to a different device. To break the dependency chain between these components, it transforms the conventional sequential denoising into an asynchronous process by exploiting the high similarity between hidden states in consecutive diffusion steps. Consequently, each component is facilitated to compute in parallel on separate devices. The proposed strategy significantly reduces inference latency while minimally impacting the generative quality. Specifically, for the Stable Diffusion v2.1, AsyncDiff achieves a 2.7x speedup with negligible degradation and a 4.0x speedup with only a slight reduction of 0.38 in CLIP Score, on four NVIDIA A5000 GPUs. Our experiments also demonstrate that AsyncDiff can be readily applied to video diffusion models with encouraging performances. The code is available at https://github.com/czg1225/AsyncDiff.

Discovering the underlying relationships among variables from temporal observations has been a longstanding challenge in numerous scientific disciplines, including biology, finance, and climate science. The dynamics of such systems are often best described using continuous-time stochastic processes. Unfortunately, most existing structure learning approaches assume that the underlying process evolves in discrete-time and/or observations occur at regular time intervals. These mismatched assumptions can often lead to incorrect learned structures and models. In this work, we introduce a novel structure learning method, SCOTCH, which combines neural stochastic differential equations (SDE) with variational inference to infer a posterior distribution over possible structures. This continuous-time approach can naturally handle both learning from and predicting observations at arbitrary time points. Theoretically, we establish sufficient conditions for an SDE and SCOTCH to be structurally identifiable, and prove its consistency under infinite data limits. Empirically, we demonstrate that our approach leads to improved structure learning performance on both synthetic and real-world datasets compared to relevant baselines under regular and irregular sampling intervals.

Diffusion probabilistic models (DPMs) are emerging powerful generative models. Despite their high-quality generation performance, DPMs still suffer from their slow sampling as they generally need hundreds or thousands of sequential function evaluations (steps) of large neural networks to draw a sample. Sampling from DPMs can be viewed alternatively as solving the corresponding diffusion ordinary differential equations (ODEs). In this work, we propose an exact formulation of the solution of diffusion ODEs. The formulation analytically computes the linear part of the solution, rather than leaving all terms to black-box ODE solvers as adopted in previous works. By applying change-of-variable, the solution can be equivalently simplified to an exponentially weighted integral of the neural network. Based on our formulation, we propose DPM-Solver, a fast dedicated high-order solver for diffusion ODEs with the convergence order guarantee. DPM-Solver is suitable for both discrete-time and continuous-time DPMs without any further training. Experimental results show that DPM-Solver can generate high-quality samples in only 10 to 20 function evaluations on various datasets. We achieve 4.70 FID in 10 function evaluations and 2.87 FID in 20 function evaluations on the CIFAR10 dataset, and a 4sim 16times speedup compared with previous state-of-the-art training-free samplers on various datasets.

Finite symmetric groups S_n are essential in fields such as combinatorics, physics, and chemistry. However, learning a probability distribution over S_n poses significant challenges due to its intractable size and discrete nature. In this paper, we introduce SymmetricDiffusers, a novel discrete diffusion model that simplifies the task of learning a complicated distribution over S_n by decomposing it into learning simpler transitions of the reverse diffusion using deep neural networks. We identify the riffle shuffle as an effective forward transition and provide empirical guidelines for selecting the diffusion length based on the theory of random walks on finite groups. Additionally, we propose a generalized Plackett-Luce (PL) distribution for the reverse transition, which is provably more expressive than the PL distribution. We further introduce a theoretically grounded "denoising schedule" to improve sampling and learning efficiency. Extensive experiments show that our model achieves state-of-the-art or comparable performances on solving tasks including sorting 4-digit MNIST images, jigsaw puzzles, and traveling salesman problems. Our code is released at https://github.com/DSL-Lab/SymmetricDiffusers.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Diffusion models, as a novel generative paradigm, have achieved remarkable success in various image generation tasks such as image inpainting, image-to-text translation, and video generation. Graph generation is a crucial computational task on graphs with numerous real-world applications. It aims to learn the distribution of given graphs and then generate new graphs. Given the great success of diffusion models in image generation, increasing efforts have been made to leverage these techniques to advance graph generation in recent years. In this paper, we first provide a comprehensive overview of generative diffusion models on graphs, In particular, we review representative algorithms for three variants of graph diffusion models, i.e., Score Matching with Langevin Dynamics (SMLD), Denoising Diffusion Probabilistic Model (DDPM), and Score-based Generative Model (SGM). Then, we summarize the major applications of generative diffusion models on graphs with a specific focus on molecule and protein modeling. Finally, we discuss promising directions in generative diffusion models on graph-structured data. For this survey, we also created a GitHub project website by collecting the supporting resources for generative diffusion models on graphs, at the link: https://github.com/ChengyiLIU-cs/Generative-Diffusion-Models-on-Graphs

We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical solvers. Most existing forward or inverse PDE approaches perform poorly when the observations on the data or the underlying coefficients are incomplete, which is a common assumption for real-world measurements. In this work, we propose DiffusionPDE that can simultaneously fill in the missing information and solve a PDE by modeling the joint distribution of the solution and coefficient spaces. We show that the learned generative priors lead to a versatile framework for accurately solving a wide range of PDEs under partial observation, significantly outperforming the state-of-the-art methods for both forward and inverse directions.

Diffusion models (DMs) demonstrate potent image generation capabilities in various generative modeling tasks. Nevertheless, their primary limitation lies in slow sampling speed, requiring hundreds or thousands of sequential function evaluations through large neural networks to generate high-quality images. Sampling from DMs can be seen alternatively as solving corresponding stochastic differential equations (SDEs) or ordinary differential equations (ODEs). In this work, we formulate the sampling process as an extended reverse-time SDE (ER SDE), unifying prior explorations into ODEs and SDEs. Leveraging the semi-linear structure of ER SDE solutions, we offer exact solutions and arbitrarily high-order approximate solutions for VP SDE and VE SDE, respectively. Based on the solution space of the ER SDE, we yield mathematical insights elucidating the superior performance of ODE solvers over SDE solvers in terms of fast sampling. Additionally, we unveil that VP SDE solvers stand on par with their VE SDE counterparts. Finally, we devise fast and training-free samplers, ER-SDE-Solvers, achieving state-of-the-art performance across all stochastic samplers. Experimental results demonstrate achieving 3.45 FID in 20 function evaluations and 2.24 FID in 50 function evaluations on the ImageNet 64times64 dataset.

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.

Standard diffusion models involve an image transform -- adding Gaussian noise -- and an image restoration operator that inverts this degradation. We observe that the generative behavior of diffusion models is not strongly dependent on the choice of image degradation, and in fact an entire family of generative models can be constructed by varying this choice. Even when using completely deterministic degradations (e.g., blur, masking, and more), the training and test-time update rules that underlie diffusion models can be easily generalized to create generative models. The success of these fully deterministic models calls into question the community's understanding of diffusion models, which relies on noise in either gradient Langevin dynamics or variational inference, and paves the way for generalized diffusion models that invert arbitrary processes. Our code is available at https://github.com/arpitbansal297/Cold-Diffusion-Models

We address causal reasoning in multivariate time series data generated by stochastic processes. Existing approaches are largely restricted to static settings, ignoring the continuity and emission of variations across time. In contrast, we propose a learning paradigm that directly establishes causation between events in the course of time. We present two key lemmas to compute causal contributions and frame them as reinforcement learning problems. Our approach offers formal and computational tools for uncovering and quantifying causal relationships in diffusion processes, subsuming various important settings such as discrete-time Markov decision processes. Finally, in fairly intricate experiments and through sheer learning, our framework reveals and quantifies causal links, which otherwise seem inexplicable.

Score-based Generative Models (SGMs) have achieved state-of-the-art synthesis results on diverse tasks. However, the current design space of the forward diffusion process is largely unexplored and often relies on physical intuition or simplifying assumptions. Leveraging results from the design of scalable Bayesian posterior samplers, we present a complete recipe for constructing forward processes in SGMs, all of which are guaranteed to converge to the target distribution of interest. We show that several existing SGMs can be cast as specific instantiations of this parameterization. Furthermore, building on this recipe, we construct a novel SGM: Phase Space Langevin Diffusion (PSLD), which performs score-based modeling in a space augmented with auxiliary variables akin to a physical phase space. We show that PSLD outperforms competing baselines in terms of sample quality and the speed-vs-quality tradeoff across different samplers on various standard image synthesis benchmarks. Moreover, we show that PSLD achieves sample quality comparable to state-of-the-art SGMs (FID: 2.10 on unconditional CIFAR-10 generation), providing an attractive alternative as an SGM backbone for further development. We will publish our code and model checkpoints for reproducibility at https://github.com/mandt-lab/PSLD.

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples -- which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

We introduce Diffusion World Model (DWM), a conditional diffusion model capable of predicting multistep future states and rewards concurrently. As opposed to traditional one-step dynamics models, DWM offers long-horizon predictions in a single forward pass, eliminating the need for recursive quires. We integrate DWM into model-based value estimation, where the short-term return is simulated by future trajectories sampled from DWM. In the context of offline reinforcement learning, DWM can be viewed as a conservative value regularization through generative modeling. Alternatively, it can be seen as a data source that enables offline Q-learning with synthetic data. Our experiments on the D4RL dataset confirm the robustness of DWM to long-horizon simulation. In terms of absolute performance, DWM significantly surpasses one-step dynamics models with a 44% performance gain, and achieves state-of-the-art performance.

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.

Representation learning is all about discovering the hidden modular attributes that generate the data faithfully. We explore the potential of Denoising Diffusion Probabilistic Model (DM) in unsupervised learning of the modular attributes. We build a theoretical framework that connects the diffusion time-steps and the hidden attributes, which serves as an effective inductive bias for unsupervised learning. Specifically, the forward diffusion process incrementally adds Gaussian noise to samples at each time-step, which essentially collapses different samples into similar ones by losing attributes, e.g., fine-grained attributes such as texture are lost with less noise added (i.e., early time-steps), while coarse-grained ones such as shape are lost by adding more noise (i.e., late time-steps). To disentangle the modular attributes, at each time-step t, we learn a t-specific feature to compensate for the newly lost attribute, and the set of all 1,...,t-specific features, corresponding to the cumulative set of lost attributes, are trained to make up for the reconstruction error of a pre-trained DM at time-step t. On CelebA, FFHQ, and Bedroom datasets, the learned feature significantly improves attribute classification and enables faithful counterfactual generation, e.g., interpolating only one specified attribute between two images, validating the disentanglement quality. Codes are in https://github.com/yue-zhongqi/diti.

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.

We present Piecewise Rectified Flow (PeRFlow), a flow-based method for accelerating diffusion models. PeRFlow divides the sampling process of generative flows into several time windows and straightens the trajectories in each interval via the reflow operation, thereby approaching piecewise linear flows. PeRFlow achieves superior performance in a few-step generation. Moreover, through dedicated parameterizations, the obtained PeRFlow models show advantageous transfer ability, serving as universal plug-and-play accelerators that are compatible with various workflows based on the pre-trained diffusion models. The implementations of training and inference are fully open-sourced. https://github.com/magic-research/piecewise-rectified-flow

We present a new algorithm to optimize distributions defined implicitly by parameterized stochastic diffusions. Doing so allows us to modify the outcome distribution of sampling processes by optimizing over their parameters. We introduce a general framework for first-order optimization of these processes, that performs jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical guarantees on the performance of our method, as well as experimental results demonstrating its effectiveness in real-world settings.

In this work, we tackle the task of point cloud denoising through a novel framework that adapts Diffusion Schr\"odinger bridges to points clouds. Unlike previous approaches that predict point-wise displacements from point features or learned noise distributions, our method learns an optimal transport plan between paired point clouds. Experiments on object datasets like PU-Net and real-world datasets such as ScanNet++ and ARKitScenes show that P2P-Bridge achieves significant improvements over existing methods. While our approach demonstrates strong results using only point coordinates, we also show that incorporating additional features, such as color information or point-wise DINOv2 features, further enhances the performance. Code and pretrained models are available at https://p2p-bridge.github.io.

Diffusion models have quickly become the state-of-the-art for generation tasks across many different data modalities. An important ability of diffusion models is the ability to encode samples from the data distribution back into the sampling prior distribution. This is useful for performing alterations to real data samples along with guided generation via the continuous adjoint equations. We propose an algebraically reversible solver for diffusion SDEs that can exactly invert real data samples into the prior distribution.

We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

Originating from the diffusion phenomenon in physics that describes particle movement, the diffusion generative models inherit the characteristics of stochastic random walk in the data space along the denoising trajectory. However, the intrinsic mutual interference among image regions contradicts the need for practical downstream application scenarios where the preservation of low-level pixel information from given conditioning is desired (e.g., customization tasks like personalized generation and inpainting based on a user-provided single image). In this work, we investigate the diffusion (physics) in diffusion (machine learning) properties and propose our Cyclic One-Way Diffusion (COW) method to control the direction of diffusion phenomenon given a pre-trained frozen diffusion model for versatile customization application scenarios, where the low-level pixel information from the conditioning needs to be preserved. Notably, unlike most current methods that incorporate additional conditions by fine-tuning the base text-to-image diffusion model or learning auxiliary networks, our method provides a novel perspective to understand the task needs and is applicable to a wider range of customization scenarios in a learning-free manner. Extensive experiment results show that our proposed COW can achieve more flexible customization based on strict visual conditions in different application settings. Project page: https://wangruoyu02.github.io/cow.github.io/.

Consistency models (CMs) are a powerful class of diffusion-based generative models optimized for fast sampling. Most existing CMs are trained using discretized timesteps, which introduce additional hyperparameters and are prone to discretization errors. While continuous-time formulations can mitigate these issues, their success has been limited by training instability. To address this, we propose a simplified theoretical framework that unifies previous parameterizations of diffusion models and CMs, identifying the root causes of instability. Based on this analysis, we introduce key improvements in diffusion process parameterization, network architecture, and training objectives. These changes enable us to train continuous-time CMs at an unprecedented scale, reaching 1.5B parameters on ImageNet 512x512. Our proposed training algorithm, using only two sampling steps, achieves FID scores of 2.06 on CIFAR-10, 1.48 on ImageNet 64x64, and 1.88 on ImageNet 512x512, narrowing the gap in FID scores with the best existing diffusion models to within 10%.

Diffusion models have emerged as the new state-of-the-art generative model with high quality samples, with intriguing properties such as mode coverage and high flexibility. They have also been shown to be effective inverse problem solvers, acting as the prior of the distribution, while the information of the forward model can be granted at the sampling stage. Nonetheless, as the generative process remains in the same high dimensional (i.e. identical to data dimension) space, the models have not been extended to 3D inverse problems due to the extremely high memory and computational cost. In this paper, we combine the ideas from the conventional model-based iterative reconstruction with the modern diffusion models, which leads to a highly effective method for solving 3D medical image reconstruction tasks such as sparse-view tomography, limited angle tomography, compressed sensing MRI from pre-trained 2D diffusion models. In essence, we propose to augment the 2D diffusion prior with a model-based prior in the remaining direction at test time, such that one can achieve coherent reconstructions across all dimensions. Our method can be run in a single commodity GPU, and establishes the new state-of-the-art, showing that the proposed method can perform reconstructions of high fidelity and accuracy even in the most extreme cases (e.g. 2-view 3D tomography). We further reveal that the generalization capacity of the proposed method is surprisingly high, and can be used to reconstruct volumes that are entirely different from the training dataset.

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

The Diffusion models, widely used for image generation, face significant challenges related to their broad applicability due to prolonged inference times and high memory demands. Efficient Post-Training Quantization (PTQ) is crucial to address these issues. However, unlike traditional models, diffusion models critically rely on the time-step for the multi-round denoising. Typically, each time-step is encoded into a hypersensitive temporal feature by several modules. Despite this, existing PTQ methods do not optimize these modules individually. Instead, they employ unsuitable reconstruction objectives and complex calibration methods, leading to significant disturbances in the temporal feature and denoising trajectory, as well as reduced compression efficiency. To address these challenges, we introduce a novel quantization framework that includes three strategies: 1) TIB-based Maintenance: Based on our innovative Temporal Information Block (TIB) definition, Temporal Information-aware Reconstruction (TIAR) and Finite Set Calibration (FSC) are developed to efficiently align original temporal features. 2) Cache-based Maintenance: Instead of indirect and complex optimization for the related modules, pre-computing and caching quantized counterparts of temporal features are developed to minimize errors. 3) Disturbance-aware Selection: Employ temporal feature errors to guide a fine-grained selection between the two maintenance strategies for further disturbance reduction. This framework preserves most of the temporal information and ensures high-quality end-to-end generation. Extensive testing on various datasets, diffusion models and hardware confirms our superior performance and acceleration..

Diffusion models achieve strong generative performance but often rely on large datasets that may include sensitive content. This challenge is compounded by the models' tendency to memorize training data, raising privacy concerns. SFBD (Lu et al., 2025) addresses this by training on corrupted data and using limited clean samples to capture local structure and improve convergence. However, its iterative denoising and fine-tuning loop requires manual coordination, making it burdensome to implement. We reinterpret SFBD as an alternating projection algorithm and introduce a continuous variant, SFBD flow, that removes the need for alternating steps. We further show its connection to consistency constraint-based methods, and demonstrate that its practical instantiation, Online SFBD, consistently outperforms strong baselines across benchmarks.

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.

Diffusion models have exhibited excellent performance in various domains. The probability flow ordinary differential equation (ODE) of diffusion models (i.e., diffusion ODEs) is a particular case of continuous normalizing flows (CNFs), which enables deterministic inference and exact likelihood evaluation. However, the likelihood estimation results by diffusion ODEs are still far from those of the state-of-the-art likelihood-based generative models. In this work, we propose several improved techniques for maximum likelihood estimation for diffusion ODEs, including both training and evaluation perspectives. For training, we propose velocity parameterization and explore variance reduction techniques for faster convergence. We also derive an error-bounded high-order flow matching objective for finetuning, which improves the ODE likelihood and smooths its trajectory. For evaluation, we propose a novel training-free truncated-normal dequantization to fill the training-evaluation gap commonly existing in diffusion ODEs. Building upon these techniques, we achieve state-of-the-art likelihood estimation results on image datasets (2.56 on CIFAR-10, 3.43/3.69 on ImageNet-32) without variational dequantization or data augmentation.

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.

Diffusion models are emerging expressive generative models, in which a large number of time steps (inference steps) are required for a single image generation. To accelerate such tedious process, reducing steps uniformly is considered as an undisputed principle of diffusion models. We consider that such a uniform assumption is not the optimal solution in practice; i.e., we can find different optimal time steps for different models. Therefore, we propose to search the optimal time steps sequence and compressed model architecture in a unified framework to achieve effective image generation for diffusion models without any further training. Specifically, we first design a unified search space that consists of all possible time steps and various architectures. Then, a two stage evolutionary algorithm is introduced to find the optimal solution in the designed search space. To further accelerate the search process, we employ FID score between generated and real samples to estimate the performance of the sampled examples. As a result, the proposed method is (i).training-free, obtaining the optimal time steps and model architecture without any training process; (ii). orthogonal to most advanced diffusion samplers and can be integrated to gain better sample quality. (iii). generalized, where the searched time steps and architectures can be directly applied on different diffusion models with the same guidance scale. Experimental results show that our method achieves excellent performance by using only a few time steps, e.g. 17.86 FID score on ImageNet 64 times 64 with only four steps, compared to 138.66 with DDIM. The code is available at https://github.com/lilijiangg/AutoDiffusion.

The optimization of the latents and parameters of diffusion models with respect to some differentiable metric defined on the output of the model is a challenging and complex problem. The sampling for diffusion models is done by solving either the probability flow ODE or diffusion SDE wherein a neural network approximates the score function allowing a numerical ODE/SDE solver to be used. However, naive backpropagation techniques are memory intensive, requiring the storage of all intermediate states, and face additional complexity in handling the injected noise from the diffusion term of the diffusion SDE. We propose a novel family of bespoke ODE solvers to the continuous adjoint equations for diffusion models, which we call AdjointDEIS. We exploit the unique construction of diffusion SDEs to further simplify the formulation of the continuous adjoint equations using exponential integrators. Moreover, we provide convergence order guarantees for our bespoke solvers. Significantly, we show that continuous adjoint equations for diffusion SDEs actually simplify to a simple ODE. Lastly, we demonstrate the effectiveness of AdjointDEIS for guided generation with an adversarial attack in the form of the face morphing problem. Our code will be released on our project page https://zblasingame.github.io/AdjointDEIS/

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.

Diffusion models have achieved great success in synthesizing high-quality images. However, generating high-resolution images with diffusion models is still challenging due to the enormous computational costs, resulting in a prohibitive latency for interactive applications. In this paper, we propose DistriFusion to tackle this problem by leveraging parallelism across multiple GPUs. Our method splits the model input into multiple patches and assigns each patch to a GPU. However, na\"{\i}vely implementing such an algorithm breaks the interaction between patches and loses fidelity, while incorporating such an interaction will incur tremendous communication overhead. To overcome this dilemma, we observe the high similarity between the input from adjacent diffusion steps and propose displaced patch parallelism, which takes advantage of the sequential nature of the diffusion process by reusing the pre-computed feature maps from the previous timestep to provide context for the current step. Therefore, our method supports asynchronous communication, which can be pipelined by computation. Extensive experiments show that our method can be applied to recent Stable Diffusion XL with no quality degradation and achieve up to a 6.1times speedup on eight NVIDIA A100s compared to one. Our code is publicly available at https://github.com/mit-han-lab/distrifuser.

Diffusion models excel in high-quality generation but suffer from slow inference due to iterative sampling. While recent methods have successfully transformed diffusion models into one-step generators, they neglect model size reduction, limiting their applicability in compute-constrained scenarios. This paper aims to develop small, efficient one-step diffusion models based on the powerful rectified flow framework, by exploring joint compression of inference steps and model size. The rectified flow framework trains one-step generative models using two operations, reflow and distillation. Compared with the original framework, squeezing the model size brings two new challenges: (1) the initialization mismatch between large teachers and small students during reflow; (2) the underperformance of naive distillation on small student models. To overcome these issues, we propose Annealing Reflow and Flow-Guided Distillation, which together comprise our SlimFlow framework. With our novel framework, we train a one-step diffusion model with an FID of 5.02 and 15.7M parameters, outperforming the previous state-of-the-art one-step diffusion model (FID=6.47, 19.4M parameters) on CIFAR10. On ImageNet 64times64 and FFHQ 64times64, our method yields small one-step diffusion models that are comparable to larger models, showcasing the effectiveness of our method in creating compact, efficient one-step diffusion models.

Human organs constantly undergo anatomical changes due to a complex mix of short-term (e.g., heartbeat) and long-term (e.g., aging) factors. Evidently, prior knowledge of these factors will be beneficial when modeling their future state, i.e., via image generation. However, most of the medical image generation tasks only rely on the input from a single image, thus ignoring the sequential dependency even when longitudinal data is available. Sequence-aware deep generative models, where model input is a sequence of ordered and timestamped images, are still underexplored in the medical imaging domain that is featured by several unique challenges: 1) Sequences with various lengths; 2) Missing data or frame, and 3) High dimensionality. To this end, we propose a sequence-aware diffusion model (SADM) for the generation of longitudinal medical images. Recently, diffusion models have shown promising results in high-fidelity image generation. Our method extends this new technique by introducing a sequence-aware transformer as the conditional module in a diffusion model. The novel design enables learning longitudinal dependency even with missing data during training and allows autoregressive generation of a sequence of images during inference. Our extensive experiments on 3D longitudinal medical images demonstrate the effectiveness of SADM compared with baselines and alternative methods. The code is available at https://github.com/ubc-tea/SADM-Longitudinal-Medical-Image-Generation.

Diffusion models based on permutation-equivariant networks can learn permutation-invariant distributions for graph data. However, in comparison to their non-invariant counterparts, we have found that these invariant models encounter greater learning challenges since 1) their effective target distributions exhibit more modes; 2) their optimal one-step denoising scores are the score functions of Gaussian mixtures with more components. Motivated by this analysis, we propose a non-invariant diffusion model, called SwinGNN, which employs an efficient edge-to-edge 2-WL message passing network and utilizes shifted window based self-attention inspired by SwinTransformers. Further, through systematic ablations, we identify several critical training and sampling techniques that significantly improve the sample quality of graph generation. At last, we introduce a simple post-processing trick, i.e., randomly permuting the generated graphs, which provably converts any graph generative model to a permutation-invariant one. Extensive experiments on synthetic and real-world protein and molecule datasets show that our SwinGNN achieves state-of-the-art performances. Our code is released at https://github.com/qiyan98/SwinGNN.

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed. The majority of such techniques consider solving the diffusion ODE due to its superior efficiency. However, stochastic sampling could offer additional advantages in generating diverse and high-quality data. In this work, we engage in a comprehensive analysis of stochastic sampling from two aspects: variance-controlled diffusion SDE and linear multi-step SDE solver. Based on our analysis, we propose SA-Solver, which is an improved efficient stochastic Adams method for solving diffusion SDE to generate data with high quality. Our experiments show that SA-Solver achieves: 1) improved or comparable performance compared with the existing state-of-the-art sampling methods for few-step sampling; 2) SOTA FID scores on substantial benchmark datasets under a suitable number of function evaluations (NFEs).

Graph generation has been dominated by autoregressive models due to their simplicity and effectiveness, despite their sensitivity to ordering. Yet diffusion models have garnered increasing attention, as they offer comparable performance while being permutation-invariant. Current graph diffusion models generate graphs in a one-shot fashion, but they require extra features and thousands of denoising steps to achieve optimal performance. We introduce PARD, a Permutation-invariant Auto Regressive Diffusion model that integrates diffusion models with autoregressive methods. PARD harnesses the effectiveness and efficiency of the autoregressive model while maintaining permutation invariance without ordering sensitivity. Specifically, we show that contrary to sets, elements in a graph are not entirely unordered and there is a unique partial order for nodes and edges. With this partial order, PARD generates a graph in a block-by-block, autoregressive fashion, where each block's probability is conditionally modeled by a shared diffusion model with an equivariant network. To ensure efficiency while being expressive, we further propose a higher-order graph transformer, which integrates transformer with PPGN. Like GPT, we extend the higher-order graph transformer to support parallel training of all blocks. Without any extra features, PARD achieves state-of-the-art performance on molecular and non-molecular datasets, and scales to large datasets like MOSES containing 1.9M molecules.

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.

Discrete diffusion has achieved state-of-the-art performance, outperforming or approaching autoregressive models on standard benchmarks. In this work, we introduce Discrete Diffusion with Planned Denoising (DDPD), a novel framework that separates the generation process into two models: a planner and a denoiser. At inference time, the planner selects which positions to denoise next by identifying the most corrupted positions in need of denoising, including both initially corrupted and those requiring additional refinement. This plan-and-denoise approach enables more efficient reconstruction during generation by iteratively identifying and denoising corruptions in the optimal order. DDPD outperforms traditional denoiser-only mask diffusion methods, achieving superior results on language modeling benchmarks such as text8, OpenWebText, and token-based generation on ImageNet 256 times 256. Notably, in language modeling, DDPD significantly reduces the performance gap between diffusion-based and autoregressive methods in terms of generative perplexity. Code is available at https://github.com/liusulin/DDPD.

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

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.

This paper presents Diffusion Forcing, a new training paradigm where a diffusion model is trained to denoise a set of tokens with independent per-token noise levels. We apply Diffusion Forcing to sequence generative modeling by training a causal next-token prediction model to generate one or several future tokens without fully diffusing past ones. Our approach is shown to combine the strengths of next-token prediction models, such as variable-length generation, with the strengths of full-sequence diffusion models, such as the ability to guide sampling to desirable trajectories. Our method offers a range of additional capabilities, such as (1) rolling-out sequences of continuous tokens, such as video, with lengths past the training horizon, where baselines diverge and (2) new sampling and guiding schemes that uniquely profit from Diffusion Forcing's variable-horizon and causal architecture, and which lead to marked performance gains in decision-making and planning tasks. In addition to its empirical success, our method is proven to optimize a variational lower bound on the likelihoods of all subsequences of tokens drawn from the true joint distribution. Project website: https://boyuan.space/diffusion-forcing/

Predicting the intermediate trajectories between an initial and target distribution is a central problem in generative modeling. Existing approaches, such as flow matching and Schr\"odinger Bridge Matching, effectively learn mappings between two distributions by modeling a single stochastic path. However, these methods are inherently limited to unimodal transitions and cannot capture branched or divergent evolution from a common origin to multiple distinct outcomes. To address this, we introduce Branched Schr\"odinger Bridge Matching (BranchSBM), a novel framework that learns branched Schr\"odinger bridges. BranchSBM parameterizes multiple time-dependent velocity fields and growth processes, enabling the representation of population-level divergence into multiple terminal distributions. We show that BranchSBM is not only more expressive but also essential for tasks involving multi-path surface navigation, modeling cell fate bifurcations from homogeneous progenitor states, and simulating diverging cellular responses to perturbations.

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.

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.

Training-free high-resolution (HR) image generation has garnered significant attention due to the high costs of training large diffusion models. Most existing methods begin by reconstructing the overall structure and then proceed to refine the local details. Despite their advancements, they still face issues with repetitive patterns in HR image generation. Besides, HR generation with diffusion models incurs significant computational costs. Thus, parallel generation is essential for interactive applications. To solve the above limitations, we introduce a novel method named ASGDiffusion for parallel HR generation with Asynchronous Structure Guidance (ASG) using pre-trained diffusion models. To solve the pattern repetition problem of HR image generation, ASGDiffusion leverages the low-resolution (LR) noise weighted by the attention mask as the structure guidance for the denoising step to ensure semantic consistency. The proposed structure guidance can significantly alleviate the pattern repetition problem. To enable parallel generation, we further propose a parallelism strategy, which calculates the patch noises and structure guidance asynchronously. By leveraging multi-GPU parallel acceleration, we significantly accelerate generation speed and reduce memory usage per GPU. Extensive experiments demonstrate that our method effectively and efficiently addresses common issues like pattern repetition and achieves state-of-the-art HR generation.

Inverse molecular design with diffusion models holds great potential for advancements in material and drug discovery. Despite success in unconditional molecular generation, integrating multiple properties such as synthetic score and gas permeability as condition constraints into diffusion models remains unexplored. We present the Graph Diffusion Transformer (Graph DiT) for multi-conditional molecular generation. Graph DiT integrates an encoder to learn numerical and categorical property representations with the Transformer-based denoiser. Unlike previous graph diffusion models that add noise separately on the atoms and bonds in the forward diffusion process, Graph DiT is trained with a novel graph-dependent noise model for accurate estimation of graph-related noise in molecules. We extensively validate Graph DiT for multi-conditional polymer and small molecule generation. Results demonstrate the superiority of Graph DiT across nine metrics from distribution learning to condition control for molecular properties. A polymer inverse design task for gas separation with feedback from domain experts further demonstrates its practical utility.

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.

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.

Discrete diffusion models with absorbing processes have shown promise in language modeling. The key quantities to be estimated are the ratios between the marginal probabilities of two transitive states at all timesteps, called the concrete score. In this paper, we reveal that the concrete score in absorbing diffusion can be expressed as conditional probabilities of clean data, multiplied by a time-dependent scalar in an analytic form. Motivated by this finding, we propose reparameterized absorbing discrete diffusion (RADD), a dedicated diffusion model without time-condition that characterizes the time-independent conditional probabilities. Besides its simplicity, RADD can reduce the number of function evaluations (NFEs) by caching the output of the time-independent network when the noisy sample remains unchanged in a sampling interval. Empirically, RADD is up to 3.5 times faster while achieving similar performance with the strongest baseline. Built upon the new perspective of conditional distributions, we further unify absorbing discrete diffusion and any-order autoregressive models (AO-ARMs), showing that the upper bound on the negative log-likelihood for the diffusion model can be interpreted as an expected negative log-likelihood for AO-ARMs. Further, our RADD models achieve SOTA performance among diffusion models on 5 zero-shot language modeling benchmarks (measured by perplexity) at the GPT-2 scale. Our code is available at https://github.com/ML-GSAI/RADD.

Diffusion-based stylization methods typically denoise from a specific partial noise state for image-to-image and video-to-video tasks. This multi-step diffusion process is computationally expensive and hinders real-world application. A promising solution to speed up the process is to obtain few-step consistency models through trajectory distillation. However, current consistency models only force the initial-step alignment between the probability flow ODE (PF-ODE) trajectories of the student and the imperfect teacher models. This training strategy can not ensure the consistency of whole trajectories. To address this issue, we propose single trajectory distillation (STD) starting from a specific partial noise state. We introduce a trajectory bank to store the teacher model's trajectory states, mitigating the time cost during training. Besides, we use an asymmetric adversarial loss to enhance the style and quality of the generated images. Extensive experiments on image and video stylization demonstrate that our method surpasses existing acceleration models in terms of style similarity and aesthetic evaluations. Our code and results will be available on the project page: https://single-trajectory-distillation.github.io.

This paper introduces Bayesian Flow Networks (BFNs), a new class of generative model in which the parameters of a set of independent distributions are modified with Bayesian inference in the light of noisy data samples, then passed as input to a neural network that outputs a second, interdependent distribution. Starting from a simple prior and iteratively updating the two distributions yields a generative procedure similar to the reverse process of diffusion models; however it is conceptually simpler in that no forward process is required. Discrete and continuous-time loss functions are derived for continuous, discretised and discrete data, along with sample generation procedures. Notably, the network inputs for discrete data lie on the probability simplex, and are therefore natively differentiable, paving the way for gradient-based sample guidance and few-step generation in discrete domains such as language modelling. The loss function directly optimises data compression and places no restrictions on the network architecture. In our experiments BFNs achieve competitive log-likelihoods for image modelling on dynamically binarized MNIST and CIFAR-10, and outperform all known discrete diffusion models on the text8 character-level language modelling task.

Generative modeling of crystal data distribution is an important yet challenging task due to the unique periodic physical symmetry of crystals. Diffusion-based methods have shown early promise in modeling crystal distribution. More recently, Bayesian Flow Networks were introduced to aggregate noisy latent variables, resulting in a variance-reduced parameter space that has been shown to be advantageous for modeling Euclidean data distributions with structural constraints (Song et al., 2023). Inspired by this, we seek to unlock its potential for modeling variables located in non-Euclidean manifolds e.g. those within crystal structures, by overcoming challenging theoretical issues. We introduce CrysBFN, a novel crystal generation method by proposing a periodic Bayesian flow, which essentially differs from the original Gaussian-based BFN by exhibiting non-monotonic entropy dynamics. To successfully realize the concept of periodic Bayesian flow, CrysBFN integrates a new entropy conditioning mechanism and empirically demonstrates its significance compared to time-conditioning. Extensive experiments over both crystal ab initio generation and crystal structure prediction tasks demonstrate the superiority of CrysBFN, which consistently achieves new state-of-the-art on all benchmarks. Surprisingly, we found that CrysBFN enjoys a significant improvement in sampling efficiency, e.g., ~100x speedup 10 v.s. 2000 steps network forwards) compared with previous diffusion-based methods on MP-20 dataset. Code is available at https://github.com/wu-han-lin/CrysBFN.

The Diffusion model, a prevalent framework for image generation, encounters significant challenges in terms of broad applicability due to its extended inference times and substantial memory requirements. Efficient Post-training Quantization (PTQ) is pivotal for addressing these issues in traditional models. Different from traditional models, diffusion models heavily depend on the time-step t to achieve satisfactory multi-round denoising. Usually, t from the finite set {1, ldots, T} is encoded to a temporal feature by a few modules totally irrespective of the sampling data. However, existing PTQ methods do not optimize these modules separately. They adopt inappropriate reconstruction targets and complex calibration methods, resulting in a severe disturbance of the temporal feature and denoising trajectory, as well as a low compression efficiency. To solve these, we propose a Temporal Feature Maintenance Quantization (TFMQ) framework building upon a Temporal Information Block which is just related to the time-step t and unrelated to the sampling data. Powered by the pioneering block design, we devise temporal information aware reconstruction (TIAR) and finite set calibration (FSC) to align the full-precision temporal features in a limited time. Equipped with the framework, we can maintain the most temporal information and ensure the end-to-end generation quality. Extensive experiments on various datasets and diffusion models prove our state-of-the-art results. Remarkably, our quantization approach, for the first time, achieves model performance nearly on par with the full-precision model under 4-bit weight quantization. Additionally, our method incurs almost no extra computational cost and accelerates quantization time by 2.0 times on LSUN-Bedrooms 256 times 256 compared to previous works.

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.

In recent advancements in high-fidelity image generation, Denoising Diffusion Probabilistic Models (DDPMs) have emerged as a key player. However, their application at high resolutions presents significant computational challenges. Current methods, such as patchifying, expedite processes in UNet and Transformer architectures but at the expense of representational capacity. Addressing this, we introduce the Diffusion State Space Model (DiffuSSM), an architecture that supplants attention mechanisms with a more scalable state space model backbone. This approach effectively handles higher resolutions without resorting to global compression, thus preserving detailed image representation throughout the diffusion process. Our focus on FLOP-efficient architectures in diffusion training marks a significant step forward. Comprehensive evaluations on both ImageNet and LSUN datasets at two resolutions demonstrate that DiffuSSMs are on par or even outperform existing diffusion models with attention modules in FID and Inception Score metrics while significantly reducing total FLOP usage.

The simulation of the metabolism in mammalian cells becomes a severe problem if spatial distributions must be taken into account. Especially the cytoplasm has a very complex geometric structure which cannot be handled by standard discretization techniques. In the present paper we propose a homogenization technique for computing effective diffusion constants. This is accomplished by using a two-step strategy. The first step consists of an analytic homogenization from the smallest to an intermediate scale. The homogenization error is estimated by comparing the analytic diffusion constant with a numerical estimate obtained by using real cell geometries. The second step consists of a random homogenization. Since no analytical solution is known to this homogenization problem, a numerical approximation algorithm is proposed. Although rather expensive this algorithm provides a reasonable estimate of the homogenized diffusion constant.

Diffusion models are a class of generative models that learn to synthesize samples by inverting a diffusion process that gradually maps data into noise. While these models have enjoyed great success recently, a full theoretical understanding of their observed properties is still lacking, in particular, their weak sensitivity to the choice of noise family and the role of adequate scheduling of noise levels for good synthesis. By identifying a correspondence between diffusion models and a well-known paradigm in cognitive science known as serial reproduction, whereby human agents iteratively observe and reproduce stimuli from memory, we show how the aforementioned properties of diffusion models can be explained as a natural consequence of this correspondence. We then complement our theoretical analysis with simulations that exhibit these key features. Our work highlights how classic paradigms in cognitive science can shed light on state-of-the-art machine learning problems.

A prominent paradigm for graph neural networks is based on the message-passing framework. In this framework, information communication is realized only between neighboring nodes. The challenge of approaches that use this paradigm is to ensure efficient and accurate long-distance communication between nodes, as deep convolutional networks are prone to oversmoothing. In this paper, we present a novel method based on time derivative graph diffusion (TIDE) to overcome these structural limitations of the message-passing framework. Our approach allows for optimizing the spatial extent of diffusion across various tasks and network channels, thus enabling medium and long-distance communication efficiently. Furthermore, we show that our architecture design also enables local message-passing and thus inherits from the capabilities of local message-passing approaches. We show that on both widely used graph benchmarks and synthetic mesh and graph datasets, the proposed framework outperforms state-of-the-art methods by a significant margin

Markov jump processes are continuous-time stochastic processes which describe dynamical systems evolving in discrete state spaces. These processes find wide application in the natural sciences and machine learning, but their inference is known to be far from trivial. In this work we introduce a methodology for zero-shot inference of Markov jump processes (MJPs), on bounded state spaces, from noisy and sparse observations, which consists of two components. First, a broad probability distribution over families of MJPs, as well as over possible observation times and noise mechanisms, with which we simulate a synthetic dataset of hidden MJPs and their noisy observation process. Second, a neural network model that processes subsets of the simulated observations, and that is trained to output the initial condition and rate matrix of the target MJP in a supervised way. We empirically demonstrate that one and the same (pretrained) model can infer, in a zero-shot fashion, hidden MJPs evolving in state spaces of different dimensionalities. Specifically, we infer MJPs which describe (i) discrete flashing ratchet systems, which are a type of Brownian motors, and the conformational dynamics in (ii) molecular simulations, (iii) experimental ion channel data and (iv) simple protein folding models. What is more, we show that our model performs on par with state-of-the-art models which are finetuned to the target datasets.

Diffusion-based image super-resolution (SR) methods are mainly limited by the low inference speed due to the requirements of hundreds or even thousands of sampling steps. Existing acceleration sampling techniques inevitably sacrifice performance to some extent, leading to over-blurry SR results. To address this issue, we propose a novel and efficient diffusion model for SR that significantly reduces the number of diffusion steps, thereby eliminating the need for post-acceleration during inference and its associated performance deterioration. Our method constructs a Markov chain that transfers between the high-resolution image and the low-resolution image by shifting the residual between them, substantially improving the transition efficiency. Additionally, an elaborate noise schedule is developed to flexibly control the shifting speed and the noise strength during the diffusion process. Extensive experiments demonstrate that the proposed method obtains superior or at least comparable performance to current state-of-the-art methods on both synthetic and real-world datasets, even only with 15 sampling steps. Our code and model are available at https://github.com/zsyOAOA/ResShift.

In this work we present a new method for the estimation of Mutual Information (MI) between random variables. Our approach is based on an original interpretation of the Girsanov theorem, which allows us to use score-based diffusion models to estimate the Kullback Leibler divergence between two densities as a difference between their score functions. As a by-product, our method also enables the estimation of the entropy of random variables. Armed with such building blocks, we present a general recipe to measure MI, which unfolds in two directions: one uses conditional diffusion process, whereas the other uses joint diffusion processes that allow simultaneous modelling of two random variables. Our results, which derive from a thorough experimental protocol over all the variants of our approach, indicate that our method is more accurate than the main alternatives from the literature, especially for challenging distributions. Furthermore, our methods pass MI self-consistency tests, including data processing and additivity under independence, which instead are a pain-point of existing methods.

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to 1 in the general case of arbitrary normalized dataset consisting of n samples in d dimension as n/drightarrow 0. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as n/drightarrow 0. We further support this theory with empirical experiments on ImageNet.

Diffusion models have shown strong competitiveness in offline reinforcement learning tasks by formulating decision-making as sequential generation. However, the practicality of these methods is limited due to the lengthy inference processes they require. In this paper, we address this problem by decomposing the sampling process of diffusion models into two decoupled subprocesses: 1) generating a feasible trajectory, which is a time-consuming process, and 2) optimizing the trajectory. With this decomposition approach, we are able to partially separate efficiency and quality factors, enabling us to simultaneously gain efficiency advantages and ensure quality assurance. We propose the Trajectory Diffuser, which utilizes a faster autoregressive model to handle the generation of feasible trajectories while retaining the trajectory optimization process of diffusion models. This allows us to achieve more efficient planning without sacrificing capability. To evaluate the effectiveness and efficiency of the Trajectory Diffuser, we conduct experiments on the D4RL benchmarks. The results demonstrate that our method achieves it 3-it 10 times faster inference speed compared to previous sequence modeling methods, while also outperforming them in terms of overall performance. https://github.com/RenMing-Huang/TrajectoryDiffuser Keywords: Reinforcement Learning and Efficient Planning and Diffusion Model

The field of image generation has made significant progress thanks to the introduction of Diffusion Models, which learn to progressively reverse a given image corruption. Recently, a few studies introduced alternative ways of corrupting images in Diffusion Models, with an emphasis on blurring. However, these studies are purely empirical and it remains unclear what is the optimal procedure for corrupting an image. In this work, we hypothesize that the optimal procedure minimizes the length of the path taken when corrupting an image towards a given final state. We propose the Fisher metric for the path length, measured in the space of probability distributions. We compute the shortest path according to this metric, and we show that it corresponds to a combination of image sharpening, rather than blurring, and noise deblurring. While the corruption was chosen arbitrarily in previous work, our Shortest Path Diffusion (SPD) determines uniquely the entire spatiotemporal structure of the corruption. We show that SPD improves on strong baselines without any hyperparameter tuning, and outperforms all previous Diffusion Models based on image blurring. Furthermore, any small deviation from the shortest path leads to worse performance, suggesting that SPD provides the optimal procedure to corrupt images. Our work sheds new light on observations made in recent works and provides a new approach to improve diffusion models on images and other types of data.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.

Diffusion models are proficient at generating high-quality images. They are however effective only when operating at the resolution used during training. Inference at a scaled resolution leads to repetitive patterns and structural distortions. Retraining at higher resolutions quickly becomes prohibitive. Thus, methods enabling pre-existing diffusion models to operate at flexible test-time resolutions are highly desirable. Previous works suffer from frequent artifacts and often introduce large latency overheads. We propose two simple modules that combine to solve these issues. We introduce a Frequency Modulation (FM) module that leverages the Fourier domain to improve the global structure consistency, and an Attention Modulation (AM) module which improves the consistency of local texture patterns, a problem largely ignored in prior works. Our method, coined Fam diffusion, can seamlessly integrate into any latent diffusion model and requires no additional training. Extensive qualitative results highlight the effectiveness of our method in addressing structural and local artifacts, while quantitative results show state-of-the-art performance. Also, our method avoids redundant inference tricks for improved consistency such as patch-based or progressive generation, leading to negligible latency overheads.

The development of video diffusion models unveils a significant challenge: the substantial computational demands. To mitigate this challenge, we note that the reverse process of diffusion exhibits an inherent entropy-reducing nature. Given the inter-frame redundancy in video modality, maintaining full frame rates in high-entropy stages is unnecessary. Based on this insight, we propose TPDiff, a unified framework to enhance training and inference efficiency. By dividing diffusion into several stages, our framework progressively increases frame rate along the diffusion process with only the last stage operating on full frame rate, thereby optimizing computational efficiency. To train the multi-stage diffusion model, we introduce a dedicated training framework: stage-wise diffusion. By solving the partitioned probability flow ordinary differential equations (ODE) of diffusion under aligned data and noise, our training strategy is applicable to various diffusion forms and further enhances training efficiency. Comprehensive experimental evaluations validate the generality of our method, demonstrating 50% reduction in training cost and 1.5x improvement in inference efficiency.

Score-based (denoising diffusion) generative models have recently gained a lot of success in generating realistic and diverse data. These approaches define a forward diffusion process for transforming data to noise and generate data by reversing it (thereby going from noise to data). Unfortunately, current score-based models generate data very slowly due to the sheer number of score network evaluations required by numerical SDE solvers. In this work, we aim to accelerate this process by devising a more efficient SDE solver. Existing approaches rely on the Euler-Maruyama (EM) solver, which uses a fixed step size. We found that naively replacing it with other SDE solvers fares poorly - they either result in low-quality samples or become slower than EM. To get around this issue, we carefully devise an SDE solver with adaptive step sizes tailored to score-based generative models piece by piece. Our solver requires only two score function evaluations, rarely rejects samples, and leads to high-quality samples. Our approach generates data 2 to 10 times faster than EM while achieving better or equal sample quality. For high-resolution images, our method leads to significantly higher quality samples than all other methods tested. Our SDE solver has the benefit of requiring no step size tuning.

Diffusion-based generative modeling has been achieving state-of-the-art results on various generation tasks. Most diffusion models, however, are limited to a single-generation modeling. Can we generalize diffusion models with the ability of multi-modal generative training for more generalizable modeling? In this paper, we propose a principled way to define a diffusion model by constructing a unified multi-modal diffusion model in a common diffusion space. We define the forward diffusion process to be driven by an information aggregation from multiple types of task-data, e.g., images for a generation task and labels for a classification task. In the reverse process, we enforce information sharing by parameterizing a shared backbone denoising network with additional modality-specific decoder heads. Such a structure can simultaneously learn to generate different types of multi-modal data with a multi-task loss, which is derived from a new multi-modal variational lower bound that generalizes the standard diffusion model. We propose several multimodal generation settings to verify our framework, including image transition, masked-image training, joint image-label and joint image-representation generative modeling. Extensive experimental results on ImageNet indicate the effectiveness of our framework for various multi-modal generative modeling, which we believe is an important research direction worthy of more future explorations.

We introduce the Fixed Point Diffusion Model (FPDM), a novel approach to image generation that integrates the concept of fixed point solving into the framework of diffusion-based generative modeling. Our approach embeds an implicit fixed point solving layer into the denoising network of a diffusion model, transforming the diffusion process into a sequence of closely-related fixed point problems. Combined with a new stochastic training method, this approach significantly reduces model size, reduces memory usage, and accelerates training. Moreover, it enables the development of two new techniques to improve sampling efficiency: reallocating computation across timesteps and reusing fixed point solutions between timesteps. We conduct extensive experiments with state-of-the-art models on ImageNet, FFHQ, CelebA-HQ, and LSUN-Church, demonstrating substantial improvements in performance and efficiency. Compared to the state-of-the-art DiT model, FPDM contains 87% fewer parameters, consumes 60% less memory during training, and improves image generation quality in situations where sampling computation or time is limited. Our code and pretrained models are available at https://lukemelas.github.io/fixed-point-diffusion-models.

A fundamental question in modern machine learning is why large, over-parameterized models, such as deep neural networks and transformers, tend to generalize well, even when their number of parameters far exceeds the number of training samples. We investigate this phenomenon through the lens of information theory, grounded in universal learning theory. Specifically, we study a Bayesian mixture learner with log-loss and (almost) uniform prior over an expansive hypothesis class. Our key result shows that the learner's regret is not determined by the overall size of the hypothesis class, but rather by the cumulative probability of all models that are close, in Kullback-Leibler divergence distance, to the true data-generating process. We refer to this cumulative probability as the weight of the hypothesis. This leads to a natural notion of model simplicity: simple models are those with large weight and thus require fewer samples to generalize, while complex models have small weight and need more data. This perspective provides a rigorous and intuitive explanation for why over-parameterized models often avoid overfitting: the presence of simple hypotheses allows the posterior to concentrate on them when supported by the data. We further bridge theory and practice by recalling that stochastic gradient descent with Langevin dynamics samples from the correct posterior distribution, enabling our theoretical learner to be approximated using standard machine learning methods combined with ensemble learning. Our analysis yields non-uniform regret bounds and aligns with key practical concepts such as flat minima and model distillation. The results apply broadly across online, batch, and supervised learning settings, offering a unified and principled understanding of the generalization behavior of modern AI systems.

Taming the generation outcome of state of the art Diffusion and Flow-Matching (FM) models without having to re-train a task-specific model unlocks a powerful tool for solving inverse problems, conditional generation, and controlled generation in general. In this work we introduce D-Flow, a simple framework for controlling the generation process by differentiating through the flow, optimizing for the source (noise) point. We motivate this framework by our key observation stating that for Diffusion/FM models trained with Gaussian probability paths, differentiating through the generation process projects gradient on the data manifold, implicitly injecting the prior into the optimization process. We validate our framework on linear and non-linear controlled generation problems including: image and audio inverse problems and conditional molecule generation reaching state of the art performance across all.

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.

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}.

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.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Diffusion models achieved great success in image synthesis, but still face challenges in high-resolution generation. Through the lens of discrete cosine transformation, we find the main reason is that the same noise level on a higher resolution results in a higher Signal-to-Noise Ratio in the frequency domain. In this work, we present Relay Diffusion Model (RDM), which transfers a low-resolution image or noise into an equivalent high-resolution one for diffusion model via blurring diffusion and block noise. Therefore, the diffusion process can continue seamlessly in any new resolution or model without restarting from pure noise or low-resolution conditioning. RDM achieves state-of-the-art FID on CelebA-HQ and sFID on ImageNet 256times256, surpassing previous works such as ADM, LDM and DiT by a large margin. All the codes and checkpoints are open-sourced at https://github.com/THUDM/RelayDiffusion.

Simulating and controlling physical systems described by partial differential equations (PDEs) are crucial tasks across science and engineering. Recently, diffusion generative models have emerged as a competitive class of methods for these tasks due to their ability to capture long-term dependencies and model high-dimensional states. However, diffusion models typically struggle with handling system states with abrupt changes and generalizing to higher resolutions. In this work, we propose Wavelet Diffusion Neural Operator (WDNO), a novel PDE simulation and control framework that enhances the handling of these complexities. WDNO comprises two key innovations. Firstly, WDNO performs diffusion-based generative modeling in the wavelet domain for the entire trajectory to handle abrupt changes and long-term dependencies effectively. Secondly, to address the issue of poor generalization across different resolutions, which is one of the fundamental tasks in modeling physical systems, we introduce multi-resolution training. We validate WDNO on five physical systems, including 1D advection equation, three challenging physical systems with abrupt changes (1D Burgers' equation, 1D compressible Navier-Stokes equation and 2D incompressible fluid), and a real-world dataset ERA5, which demonstrates superior performance on both simulation and control tasks over state-of-the-art methods, with significant improvements in long-term and detail prediction accuracy. Remarkably, in the challenging context of the 2D high-dimensional and indirect control task aimed at reducing smoke leakage, WDNO reduces the leakage by 33.2% compared to the second-best baseline. The code can be found at https://github.com/AI4Science-WestlakeU/wdno.git.

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.

Diffusion model-based approaches have shown promise in data-driven planning, but there are no safety guarantees, thus making it hard to be applied for safety-critical applications. To address these challenges, we propose a new method, called SafeDiffuser, to ensure diffusion probabilistic models satisfy specifications by using a class of control barrier functions. The key idea of our approach is to embed the proposed finite-time diffusion invariance into the denoising diffusion procedure, which enables trustworthy diffusion data generation. Moreover, we demonstrate that our finite-time diffusion invariance method through generative models not only maintains generalization performance but also creates robustness in safe data generation. We test our method on a series of safe planning tasks, including maze path generation, legged robot locomotion, and 3D space manipulation, with results showing the advantages of robustness and guarantees over vanilla diffusion models.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.