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Cited by 1,704Generative Adversarial Networks and Image SynthesisAdversarial Robustness in Machine LearningImage and Signal Denoising Methods

Diffusion Models: A Comprehensive Survey of Methods and Applications

Open access
Yang, Ling, Zhang, Zhilong, Song, Yang, Hong, Shenda, Xu, Runsheng, Zhao, Yue, Zhang, Wentao, Cui, Bin +1 more·2024·ACM Computing Surveys
DOI 10.1145/3626235
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Abstract

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

AI snapshot

Written from the paper’s full text.

TL;DR

This survey organizes diffusion-model research into three core areas—efficient sampling, improved likelihood estimation, and handling data with special structures—and reviews connections to other generative models plus six application domains. It also introduces DDPMs, score-based generative models, and Score SDEs as the main formulations and outlines future research directions.

Key findings

  • The paper proposes a taxonomy of diffusion-model advances: efficient sampling, improved likelihood estimation, and methods for data with special structures such as discrete data, invariant structures, and manifold structures.
  • Efficient sampling methods are split into learning-free and learning-based approaches; customized ODE solvers such as DPM-solver and exponential integrators can produce high-quality samples in just 10 to 20 iterations, far fewer than the hundreds typically required without accelerated sampling.
  • Improved-likelihood methods include noise schedule optimization, reverse variance learning, and exact likelihood computation; Variational Diffusion Models show that noise schedules do not affect the VLB as long as they share the same values at \(R_{\min}\) and \(R_{\max}\), and only affect Monte Carlo estimator variance.
  • Applications are grouped into six categories: computer vision, natural language processing, temporal data modeling, multi-modal learning, robust learning, and interdisciplinary applications.

Methods

  • Design: a comprehensive survey/overview of diffusion models, not an empirical study with new experiments.
  • Data/sample: existing published research literature on diffusion models and related generative models, cited throughout the reference list.
  • Main analysis: conceptual and taxonomic synthesis, including a self-contained introduction to DDPMs, SGMs, and Score SDEs, followed by categorized review of methods, connections, applications, and future directions.

Limitations

  • Stated: the paper identifies open issues including revisiting assumptions such as complete information erasure in finite time, lack of theoretical understanding for why and when diffusion models work, weaker latent representations than VAEs/GANs, and pitfalls such as dataset bias, improper image generation, and privacy issues.
  • Stated: for scene-graph-to-image generation, high-quality paired SG-image datasets are scarce, and leveraging large-scale text-image datasets for augmentation or semantic diffusion priors remains an open problem.
  • Evident: the survey does not report a systematic search protocol, inclusion/exclusion criteria, or quantitative meta-analysis, so its coverage is broad but potentially selective.

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Integrity signals

Registry

Crossref · 10.1145/3626235

Cites

236 of 236 references verified0 citation errors · 0 not checkable

Statistics

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Claims the paper makes

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  • Premisecitation

    Diffusion models have broken the long-time dominance of generative adversarial networks in image synthesis.

    “They have broken the long-time dominance of generative adversarial networks (GANs) [71] in the challenging task of image synthesis [54,87,223,228]”
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    Diffusion models have emerged as the new state-of-the-art family of deep generative models.

    “Diffusion models [87,218,223,228] have emerged as the new state-of-the-art family of deep generative models.”
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  • Premisereasoning

    Key to the success of DDPM sampling is training the reverse Markov chain to match the actual time reversal of the forward Markov chain.

    “Key to the success of this sampling process is training the reverse Markov chain to match the actual time reversal of the forward Markov chain.”
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    Numerous methods have been developed to improve diffusion models by enhancing empirical performance or extending model capacity from a theoretical perspective.

    “Numerous methods have been developed to improve diffusion models, either by enhancing empirical performance [165,220,224] or by extending the model capacity from a theoretical perspective [144,145,222,228,279].”
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  • Premisenone given

    Key to the three predominant diffusion formulations is to progressively perturb data with intensifying random noise and then successively remove noise to generate new data samples.

    “Key to all these approaches is to progressively perturb data with intensifying random noise (called the “diffusion” process), then successively remove noise to generate new data samples.”
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    Premisecitation

    Diffusion models have shown potential in computer vision, natural language processing, temporal data modeling, multi-modal modeling, robust machine learning, computational chemistry, and medical image reconstruction.

    “and have also shown potential in a variety of domains, ranging from computer vision [2,11,19,23,88,90,113,115,134,150,160,173,198,200,248,270,271,283,290], natural language processing [6,93,139,205,275], temporal data modeling [1,33,124,191,233,262], multi-modal modeling [7,186,196,199,288], robust machine learning [18,28,112,242,273], to interdisciplinary applications in fields such as computa”
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    Interpretationnone given

    The three predominant formulations-DDPMs, SGMs, and Score SDEs-are connected and can be reduced to one another.

    “We clarify how they work under the same principle of diffusion and explain how these three models are connected and can be reduced to one another.”
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    Premisecitation

    Ho et al. propose to reweight terms in the variational lower bound for better sample quality and noticed an equivalence between the resulting loss and the training objective for noise-conditional score networks.

    “Ho et al. (2020) [87] propose to reweight various terms in L VLB for better sample quality and noticed an important equivalence between the resulting loss function and the training objective for noise-conditional score networks (NCSNs), one type of score-based generative models, in Song and Ermon (2019) [223].”
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    Interpretationcitation

    The DDPM loss reduces to Equation (8) for a particular choice of the weighting function λ(t) and has the same form as the denoising score matching loss for score-based generative models.

    “This objective reduces to Equation (8) for a particular choice of the weighting function λ(t ), and has the same form as the loss of denoising score matching over multiple noise scales for training score-based generative models [223]”
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    Premisereasoning

    Training and sampling are completely decoupled in score-based generative models, so many sampling techniques can be used after score estimation.

    “Training and sampling are completely decoupled in the formulation of score-based generative models, so one can use a multitude of sampling techniques after the estimation of score functions.”
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    Interpretationreasoning

    Both the reverse-time SDE and the probability flow ODE allow sampling from the same data distribution as their trajectories have the same marginals.

    “Both the reverse-time SDE and the probability flow ODE allow sampling from the same data distribution as their trajectories have the same marginals.”
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    Premisecitation

    The added velocity term in CLD is reported to improve sampling speed as well as quality.

    “The added velocity term is reported to improve sampling speed as well as quality.”
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    Premisecitation

    NCSNs and CLD both solve the reverse-time SDE with inspirations from Langevin dynamics.

    “Noise-Conditional Score Networks (NCSNs) [223] and Critically-Damped Langevin Diffusion (CLD) [59] both solve the reverse-time SDE with inspirations from Langevin dynamics.”
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    The forward processes in DDPMs and SGMs are both discretizations of the Score SDE.

    “The forward processes in DDPMs and SGMs are both discretizations of this SDE.”
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    Premisenone given

    Generating samples from diffusion models typically demands iterative approaches involving a large number of evaluation steps.

    “Generating samples from diffusion models typically demands iterative approaches that involve a large number of evaluation steps.”
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    Premisecitation

    CLD proposes an augmented SDE with an auxiliary velocity term resembling underdamped Langevin diffusion, and to obtain the time reversal it only needs to learn the score function of the conditional distribution of velocity given data, arguably easier than learning scores of data directly.

    “Inspired by statistical mechanics, CLD proposes an augmented SDE with an auxiliary velocity term resembling underdamped Langevin diffusion.To obtain the time reversal of the extended SDE, CLD only needs to learn the score function of the conditional distribution of velocity given data, arguably easier than learning scores of data directly.”
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    The reverse diffusion process can be applied to any type of forward SDE, and empirical results indicate it performs slightly better than DDPM for the VP-SDE.

    “This process can be applied to any types of forward SDEs, and empirical results indicate this sampler performs slightly better than DDPM [228] for a particular type of SDEs called the VP-SDE.”
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    Subsequent research on diffusion models focuses on improving classical approaches from three major directions: faster and more efficient sampling, more accurate likelihood and density estimation, and handling data with special structures.

    “Subsequent research on diffusion models focuses on improving these classical approaches (DDPMs, SGMs, and Score SDEs) from three major directions: faster and more efficient sampling, more accurate likelihood and density estimation, and handling data with special structures (such as permutation invariance, manifold structures, and discrete data).”
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    Jolicoeur-Martineau et al. developed an SDE solver with adaptive step sizes for faster generation.

    “Jolicoeur-Martineau et al. (2021) [107] develop an SDE solver with adaptive step sizes for faster generation.”
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    ALD sampling trajectories are not exact solutions to the reverse-time SDE but have the correct marginals and produce correct samples under the assumption that Langevin dynamics converges to equilibrium at every noise level.

    “Although the sampling trajectories of ALD are not exact solutions to the reverse-time SDE, they have the correct marginals and hence produce correct samples under the assumption that Langevin dynamics converges to its equilibrium at every noise level.”
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    Premisecitation

    PNDM proposes a pseudo numerical method to generate samples along a specific manifold and encapsulates DDIM as a special case.

    “PNDM [142] proposes a pseudo numerical method to generate sample along a specific manifold in R N .It uses numerical solver with nonlinear transfer part to solve differential equation on manifolds and then generates sample, which encapsulates DDIM as a special case.”
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    Karras et al. improved the Langevin dynamics corrector by proposing a Langevin-like churn step of adding and removing noise, achieving new state-of-the-art sample quality on datasets like CIFAR-10 and ImageNet-64.

    “Karras et al. (2022) [111] further improve the Langevin dynamics corrector in [228] by proposing a Langevin-like “churn” step of adding and removing noise, achieving new state-of-the-art sample quality on datasets like CIFAR-10 [126] and ImageNet-64 [53].”
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    Premisecitation

    DDIM learns a Markov chain to reverse the non-Markov perturbation process, which is fully deterministic when σ2_t = 0.

    “DDIM learns a Markov chain to reverse this non-Markov perturbation process, which is fully deterministic when σ 2 t = 0.”
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    It has been observed that the DDIM sampling process amounts to a special discretization scheme of the probability flow ODE.

    “It is observed in [111,145,201,220] that the DDIM sampling process amounts to a special discretization scheme of the probability flow ODE.”
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    Premisecitation

    The predictor-corrector method solves the reverse SDE by combining numerical SDE solvers and iterative MCMC, and its resulting samples have the same time-marginals as solution trajectories of the reverse-time SDE.

    “The predictor-corrector method proposed in [228] solves the reverse SDE by combining numerical SDE solvers (“predictor”) and iterative Markov chain Monte Carlo (MCMC) approaches (“corrector”).”
    • Also stated in: § 3.1.1 SDE Solvers. · plain
      “The resulting samples have the same time-marginals as solution trajectories of the reverse-time SDE, i.e., they are equivalent in distribution at all time steps.”
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    Premisenone given

    In contrast to SDE solvers, the trajectories of ODE solvers are deterministic and thus not affected by stochastic fluctuations.

    “In contrast to SDE solvers, the trajectories of ODE solvers are deterministic and thus not affected by stochastic fluctuations.”
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    Premisecitation

    Dockhorn et al. derived a second-order solver for accelerating synthesis by training an additional head on top of the first-order score network.

    “Based on truncated Taylor methods, Dockhorn et al. (2022) [60] derive a second-order solver for accelerating synthesis by training an additional head on top of the first-order score network.”
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    Premisecitation

    Watson et al. put forth a strategy for finding the optimal discretization scheme by selecting the best K time steps to maximize the DDPM training objective.

    “Given a pre-trained diffusion model, Watson et al. (2021) [247] put forth a strategy for finding the optimal discretization scheme by selecting the best K time steps to maximize the training objective for DDPMs.”
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    Premisenone given

    The DDPM objective can be broken down into a sum of individual terms, making it well suited for dynamic programming.

    “Key to this approach is the observation that the DDPM objective can be broken down into a sum of individual terms, making it well suited for dynamic programming.”
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    Premisecitation

    These customized ODE methods contain DDIM as a first-order approximation and allow higher-order integrators that can produce high-quality samples in just 10 to 20 iterations, far fewer than the hundreds typically required without accelerated sampling.

    “These methods contain DDIM as a first-order approximation.However, they also allow for higher order integrators, which can produce highquality samples in just 10 to 20 iterations-far fewer than the hundreds of iterations typically required by diffusion models without accelerated sampling.”
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    Premisecitation

    Repeating progressive distillation can further reduce sampling steps, although fewer steps can result in reduced sample quality.

    “Repeating this procedure can further reduce sampling steps, although fewer steps can result in reduced sample quality.”
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    Premisecitation

    The variational lower bound may not be tight in many cases, leading to potentially suboptimal log-likelihoods from diffusion models.

    “This bound, however, may not be tight in many cases [120], leading to potentially suboptimal log-likelihoods from diffusion models.”
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    Interpretationreasoning

    Noise schedules do not affect the variational lower bound as long as they share the same values at R_min and R_max, and they only affect the variance of Monte Carlo estimators for the variational lower bound.

    “As a result, noise schedules do not affect the VLB as long as they share the same values at R min and R max , and will only affect the variance of Monte Carlo estimators for VLB.”
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    Interpretationcitation

    Given a pre-trained score model, estimating its first- and second-order moments yields the optimal reverse variances, and plugging them into the variational lower bound can lead to tighter bounds and higher likelihood values.

    “As a result, given a pre-trained score model, we can estimate its first-and second-order moments to obtain the optimal reverse variances.Plugging them into the VLB can lead to tighter VLBs and higher likelihood values.”
    Quote in the paperSubstantial claim

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    Premisecitation

    iDDPM learns the reverse variances by parameterizing them with linear interpolation and training them using a hybrid objective, resulting in higher log-likelihoods and faster sampling without losing sample quality.

    “In iDDPM [165], Nichol and Dhariwal propose to learn the reverse variances by parameterizing them with a form of linear interpolation and training them using a hybrid objective.This results in higher log-likelihoods and faster sampling without losing sample quality.”
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    The classical formulation of diffusion models assumes that Gaussian transition kernels in the reverse Markov chain have fixed variance parameters.

    “The classical formulation of diffusion models assumes that Gaussian transition kernels in the reverse Markov chain have fixed variance parameters.”
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    Premisecitation

    For p_sde, several concurrent works demonstrate that there exists an efficiently computable variational lower bound, and diffusion models can be directly trained to maximize p_sde using modified diffusion losses.

    “For p sde θ , several concurrent works [95,144,222] demonstrate that there exists an efficiently computable variational lower bound, and we can directly train our diffusion models to maximize p sde θ using modified diffusion losses.”
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    Interpretationreasoning

    Since the probability flow ODE is a special case of neural ODEs or continuous normalizing flows, well-established approaches in those fields can compute log p_ode accurately.

    “Since the probability flow ODE is a special case of neural ODEs or continuous normalizing flows, we can use well-established approaches in those fields to compute log p ode θ accurately.”
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    Premisenone given

    Difficulties may arise when models rely on score functions that are only defined on continuous data domains, or when data reside on low dimensional manifolds.

    “Difficulties may arise, for example, when models rely on score functions that are only defined on continuous data domains, or when data reside on low dimensional manifolds.”
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    Premisereasoning

    Most diffusion models are geared towards continuous data domains because Gaussian noise perturbation is not a natural fit for discrete data, and the score functions required by SGMs and Score SDEs are only defined on continuous data domains.

    “Most diffusion models are geared towards continuous data domains, because Gaussian noise perturbation as used in DDPMs is not a natural fit for discrete data, and the score functions required by SGMs and Score SDEs are only defined on continuous data domains.”
    Quote in the paperSubstantial claim

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    Premisecitation

    Leveraging Continuous Time Markov Chains, Campbell et al. derive efficient samplers that outperform discrete counterparts, while providing a theoretical analysis on the error between the sample distribution and the true data distribution.

    “Leveraging Continuous Time Markov Chains, they are able to derive efficient samplers that outperform discrete counterparts, while providing a theoretical analysis on the error between the sample distribution and the true data distribution.”
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    Premisecitation

    VQ-Diffusion replaces Gaussian noise with a random walk on the discrete data space, or a random masking operation.

    “Specifically, VQ-Diffusion [78] replaces Gaussian noise with a random walk on the discrete data space, or a random masking operation.”
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    Premisecitation

    Lu et al. propose efficient training algorithms for minimizing high order score matching losses and reported improved p_ode on data.

    “Building upon this theoretical result, authors further propose efficient training algorithms for minimizing high order score matching losses and reported improved p ode θ on data.”
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    Premisereasoning

    Diffusion models do not necessarily translate seamlessly to other modalities, and many important data domains have special structures that must be taken into account for diffusion models to function effectively.

    “While diffusion models have achieved great success for data domains like images and audio, they do not necessarily translate seamlessly to other modalities.Many important data domains have special structures that must be taken into account for diffusion models to function effectively.”
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    Premisenone given

    Data in many important domains have invariant structures; graphs are permutation invariant, and point clouds are both translation and rotation invariant.

    “Data in many important domains have invariant structures.For example, graphs are permutation invariant, and point clouds are both translation and rotation invariant.”
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    Premisecitation

    Xu et al. (2022) shows that Markov chains starting with an invariant prior and evolving with equivariant Markov kernels can induce an invariant marginal distribution, which can be used to enforce appropriate data invariance in molecular conformation generation.

    “For example, Xu et al. (2022) [261] shows that Markov chains starting with an invariant prior and evolving with equivariant Markov kernels can induce an invariant marginal distribution, which can be used to enforce appropriate data invariance in molecular conformation generation.”
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    Premisecitation

    GDSS models both the joint distribution of nodes and edges through a system of SDEs, where message-passing operations guarantee permutation invariance.

    “This process models both the joint distribution of nodes and edges through a system of stochastic differential equations (SDEs), where message-passing operations are used to guarantee permutation invariance.”
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    Premisenone given

    In diffusion models, these invariances are often ignored, which can lead to suboptimal performance.

    “In diffusion models, these invariances are often ignored, which can lead to suboptimal performance.”
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    GDSS further develops this idea by proposing a continuous-time graph diffusion process.

    “GDSS [105] further develops this idea by proposing a continuous-time graph diffusion process.”
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    Interpretationreasoning

    In order to be successful, it is crucial to design a loss function that allows for the joint training of the autoencoder and the diffusion models.

    “In order to be successful, it is crucial to design a loss function that allows for the joint training of the autoencoder and the diffusion models.”
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    Premisecitation

    Unlike RSGM, RDM takes an extrinsic view, assuming that the relevant Riemannian manifold is embedded in a higher dimensional Euclidean space.

    “Unlike the RSGM, the RDM takes an extrinsic view, assuming that the relevant Riemannian manifold is embedded in a higher dimensional Euclidean space.”
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    Premisereasoning

    Many recent works have built on this idea, starting by using an autoencoder to condense the data into a lower dimensional manifold, followed by training diffusion models in this latent space.

    “Many recent works have built on this idea, starting by using an autoencoder to condense the data into a lower dimensional manifold, followed by training diffusion models in this latent space.”
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    Premisecitation

    The authors of RDM have shown that maximizing this VLB is equivalent to minimizing a Riemannian score matching loss.

    “The authors of the RDM model have shown that maximizing this VLB is equivalent to minimizing a Riemannian scorematching loss.”
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    Premisereasoning

    Many data domains have well-known manifold structures; climate and earth data naturally lie on the sphere because that is the shape of our planet.

    “In addition, many data domains have well-known manifold structures.For instance, climate and earth data naturally lie on the sphere because that is the shape of our planet.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    According to the manifold hypothesis, most natural data lies on manifolds with significantly reduced intrinsic dimensionality.

    “According to the manifold hypothesis [66], most natural data lies on manifolds with significantly reduced intrinsic dimensionality.”
    Quote in the paperSubstantial claim0 of 1 sources checked
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    Premisecitation

    This adaptation parallels the generalization of neural ODEs and continuous normalizing flows to Riemannian manifolds.

    “This adaptation parallels the generalization of neural ODEs [34] and continuous normalizing flows [74] to Riemannian manifolds [143,157].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
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    Premisecitation

    Data with manifold structures are ubiquitous in machine learning; the manifold hypothesis posits that natural data often reside on manifolds with lower intrinsic dimensionality.

    “Data with manifold structures are ubiquitous in machine learning.As the manifold hypothesis [66] posits, natural data often reside on manifolds with lower intrinsic dimensionality.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [66] source

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    Premisecitation

    RSGM is trained with a generalized denoising score matching objective.

    “It is trained with a generalized denoising score matching objective.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [51]

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    Premisereasoning

    In these cases, the manifold is implicitly defined by the autoencoder and learned through the reconstruction loss.

    “In these cases, the manifold is implicitly defined by the autoencoder and learned through the reconstruction loss.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisereasoning

    This formulation allows flexible choice of encoder and decoder models.

    “This formulation allows flexible choice of encoder and decoder models.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given

    Typically, VAE models are represented by exponential family distributions whose parameters are generated by multi-layer neural networks.

    “Typically, these models are represented by exponential family distributions whose parameters are generated by multi-layer neural networks.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Variational Autoencoders aim to learn both an encoder and a decoder to map input data to values in a continuous latent space.

    “Variational Autoencoders [61,123,194] aim to learn both an encoder and a decoder to map input data to values in a continuous latent space.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
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  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    LDM first trains an autoencoder to produce a low-dimensional latent space, then trains the diffusion model to generate latent codes.

    “First, an autoencoder is trained to produce a low-dimensional latent space.Then, the diffusion model is trained to generate latent codes.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisenone given

    DDPM's forward process functions as the encoder, and this process is structured as a linear Gaussian model.

    “Specifically, DDPM's forward process functions as the encoder, and this process is structured as a linear Gaussian model (as described by Equation ( 2)).”
    Quote in the paperNot a substantial claim

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    Premisenone given

    To guarantee effective inference, a variational Bayes approach is used to maximize the evidence lower bound.

    “To guarantee an effective inference, a variational Bayes approach is used to maximize the evidence lower bound:”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    The latent variables within the DDPM decoder are all the same size as the sample data.

    “The latent variables within the decoder are all the same size as the sample data.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    DALLE-2 employs a similar strategy by training a diffusion model on the CLIP image embedding space, followed by training a separate decoder to create images based on the CLIP image embeddings.

    “DALLE-2 [186] employs a similar strategy by training a diffusion model on the CLIP image embedding space, followed by training a separate decoder to create images based on the CLIP image embeddings.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Interpretationcitation

    Consequently, optimizing a diffusion model can be seen as training an infinitely deep hierarchical VAE, supporting the common belief that Score SDE diffusion models can be interpreted as the continuous limit of hierarchical VAEs.

    “Consequently, optimizing a diffusion model can be seen as training an infinitely deep hierarchical VAE-a finding that supports the common belief that Score SDE diffusion models can be interpreted as the continuous limit of hierarchical VAEs.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisenone given

    DDPM's reverse process corresponds to the decoder, which is shared across multiple decoding steps.

    “The DDPM's reverse process, on the other hand, corresponds to the decoder, which is shared across multiple decoding steps.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    Provided that the parameterized likelihood function and posterior approximation can be computed point-wise and are differentiable with their parameters, the ELBO can be maximized with gradient descent.

    “Provided that the parameterized likelihood function p θ (x | z) and the parameterized posterior approximation q ϕ (z | x) can be computed in a point-wise way and are differentiable with their parameters, the ELBO can be maximized with gradient descent.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisenone given

    In VAE, the embedding can be interpreted as a latent variable in a probabilistic generative model, and a probabilistic decoder can be formulated by a parameterized likelihood function.

    “In these models, the embedding can be interpreted as a latent variable in a probabilistic generative model, and a probabilistic decoder can be formulated by a parameterized likelihood function.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The goal of GAN optimization is to achieve Nash equilibrium.

    “Namely, the goal of GAN optimization is to achieve Nash equilibrium [192].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Wang et al. (2022) inject noise to the discriminator with an adaptive noise schedule determined by a diffusion model.

    “Taking advantage of the flexible diffusion model, Wang et al. (2022) [245] inject noise to the discriminator with an adaptive noise schedule determined by a diffusion model.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisenone given

    The optimization of GANs can be viewed as a minimax optimization problem with value function V(G,D).

    “The optimization of GANs can be viewed as a minimax optimization problem with value function V (G, D):”
    Quote in the paperNot a substantial claim

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    Premisecitation

    Xiao et al. (2021) show that slow sampling is caused by the Gaussian assumption in the denoising step, which is justified only for small step sizes.

    “Xiao et al. (2021) [256] show that slow sampling is caused by the Gaussian assumption in the denoising step, which is justified only for small step sizes.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisenone given

    The generator G aims to generate new examples and implicitly model the data distribution.

    “The generator G aims to generate new examples and implicitly model the data distribution.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    One solution is to inject noise into the discriminator input for widening the support of both the generator and discriminator distributions.

    “One solution is to inject noise into the discriminator input for widening the support of both the generator and discriminator distributions.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisenone given

    Normalizing flows can transform a simple probability distribution into an extremely complex probability distribution, which can be used in generative models, reinforcement learning, variational inference, and other fields.

    “Normalizing flows can transform simple probability distribution into an extremely complex probability distribution, which can be used in generative models, reinforcement learning, variational inference, and other fields.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    As such, each denoising step is modeled by a conditional GAN, allowing larger step size.

    “As such, each denoising step is modeled by a conditional GAN, allowing larger step size.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisereasoning

    At Nash equilibrium, the generator can be considered to have captured the accurate distribution of real examples.

    “At that point, the generator can be considered that it has captured the accurate distribution of real examples.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    GANs mainly consist of two models: a generator G and a discriminator D.

    “Generative Adversarial Networks (GANs) [45,71,79] mainly consist of two models: a generator G and a discriminator D.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
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    Premisenone given

    The discriminator D is usually a binary classifier used to identify generated examples from true examples with maximally possible accuracy.

    “The discriminator D is usually a binary classifier that is used to identify generated examples from true examples with maximally possible accuracy.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Normalizing flows are generative models that generate tractable distributions to model high-dimensional data.

    “Normalizing flows [56,193] are generative models that generate tractable distributions to model high-dimensional data [57,121].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
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    Premisenone given

    One issue of GAN is instability in the training process, mainly caused by non-overlapping between the distribution of input data and that of the generated data.

    “One of the issues of GAN is the instability in the training process, which is mainly caused by the non-overlapping between the distribution of input data and that of the generated data.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    The optimization process ends at a saddle point that produces a minimum about the generator and a maximum about the discriminator.

    “The optimization process ends at a saddle point that produces a minimum about the generator and a maximum about the discriminator.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given

    The trajectory in normalizing flows is formulated by a differential equation.

    “The trajectory in normalizing flows is formulated by a differential equation.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    Normalizing flows allow for the retrieval of the exact log-likelihood through a change of variable formula.

    “Similar to the continuous setting, normalizing flows allow for the retrieval of the exact loglikelihood through a change of variable formula.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    DiffFlow introduces a generative modeling algorithm that combines the benefits of both flow-based and diffusion models.

    “For example, DiffFlow [278] introduces a generative modeling algorithm that combines the benefits of both flow-based and diffusion models.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Autoregressive Models work by decomposing the joint distribution of data into a product of conditional distributions using the probability chain rule.

    “Autoregressive Models (ARMs) work by decomposing the joint distribution of data into a product of conditional distributions using the probability chain rule:”
    Quote in the paperNot a substantial claim0 of 2 sources checked
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    Premisecitation

    Existing normalizing flows are constructed based on the change of variable formula.

    “Existing normalizing flows are constructed based on the change of variable formula [56,193].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [56]

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    Premisecitation

    DiffFlow produces sharper boundaries than normalizing flow and learns more general distributions with fewer discretization steps compared to diffusion probabilistic models.

    “As a result, DiffFlow produces sharper boundaries than normalizing flow and learns more general distributions with fewer discretization steps compared to diffusion probabilistic models.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Several works attempt to relax the bijection requirement.

    “Several works attempt to relax this bijection requirement [57,249].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
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    Premisecitation

    Recent advances in deep learning have facilitated significant progress for various data modalities such as images, audio, and text.

    “Recent advances in deep learning have facilitated significant progress for various data modalities [29,161,205], such as images [38,239], audio [110,170], and text [14,20,75,159,162].”
    Quote in the paperNot a substantial claim0 of 10 sources checked
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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    In the discrete-time setting, the mapping from data x to latent z in normalizing flows is a composition of a sequence of bijections.

    “In the discrete-time setting, the mapping from data x to latent z in normalizing flows is a composition of a sequence of bijections, taking the form of F = F N • F N -1 • . . .• F 1 .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisenone given

    An energy-based model is defined as p_theta(x) = 1/Z_theta exp(f_theta(x)), where Z_theta is the partition function, which is analytically intractable for high-dimensional x.

    “An energy-based model is defined as p θ (x) = 1 Z θ exp( f θ (x)), where Z θ = exp( f θ (x))dx is the partition function, which is analytically intractable for high-dimensional x.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisenone given

    Learning EBMs by maximizing the likelihood requires MCMC method to generate samples from the model, which can be very computationally expensive.

    “First, learning EBMs by maximizing the likelihood requires MCMC method to generate samples from the model, which can be very computationally expensive.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    Sampling from autoregressive models requires the same number of network calls as the data's dimensionality.

    “Sampling from these models requires the same number of network calls as the data's dimensionality.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    The Autoregressive Diffusion Model is capable of generating arbitrary-order data, including order-agnostic autoregressive models and discrete diffusion models as special cases.

    “The Autoregressive Diffusion Model (ARDM) [92], on the other hand, is capable of generating arbitrary-order data, including order-agnostic autoregressive models and discrete diffusion models as special cases [6,93,219].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
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    • cites [219]

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  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Energy-based Models can be viewed as one generative version of discriminators, while can be learned from unlabeled input data.

    “Energy-based Models (EBMs) [119,129,226] can be viewed as one generative version of discriminators [101,128,131], while can be learned from unlabeled input data.”
    Quote in the paperNot a substantial claim0 of 6 sources checked
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  • May combine two claims.

    Premisereasoning

    While autoregressive models are effective density estimators, sampling is a continuous, time-consuming process, particularly for high-dimensional data.

    “While ARMs are effective density estimators, sampling is a continuous, time-consuming process-particularly for high-dimensional data.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    For images, f_theta(x) is parameterized by a convolutional neural network with a scalar output.

    “For images, f θ (x) is parameterized by a convolutional neural network with a scalar output.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    As demonstrated in [168], the energy potentials learned with non-convergent MCMC are not stable, in the sense that samples from long-run Markov chains can be significantly different from the observed samples, and thus it is difficult to evaluate the learned energy potentials.

    “Second, as demonstrated in [168], the energy potentials learned with non-convergent MCMC are not stable, in the sense that samples from long-run Markov chains can be significantly different from the observed samples, and thus it is difficult to evaluate the learned energy potentials.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [168]

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    Premisecitation

    Salimans and Ho (2021) compare constrained score models and energy-based models for modeling the score of the data distribution, finding that constrained score models, i.e., energy based models, can perform just as well as unconstrained models when using a comparable model structure.

    “Salimans and Ho (2021) [202] compare both constrained score models and energy-based models for modeling the score of the data distribution, finding that constrained score models, i.e., energy based models, can perform just as well as unconstrained models when using a comparable model structure.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [202]

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  • May combine two claims.

    Premisecitation

    At the testing stage, the ARDM is able to generate data in parallel, enabling its application to a range of arbitrary-generation tasks.

    “At the testing stage, the ARDM is able to generate data in parallel-enabling its application to a range of arbitrary-generation tasks.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    Autoregressive models offer generative capabilities through the use of a single neural network.

    “Autoregressive models (ARMs) offer generative capabilities through the use of a single neural network.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The Flexible Diffusion Model uses a generative model to allow for the sampling of any arbitrary subset of video frames, given any other subset.

    “For example, the Flexible Diffusion Model (FDM) [83] uses a generative model to allow for the sampling of any arbitrary subset of video frames, given any other subset.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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  • May combine two claims.

    Premisereasoning

    EBMs maximize the recovery likelihood because it is more tractable than marginal likelihood, as sampling from the conditional distributions is much easier than sampling from the marginal distributions.

    “EBMs maximize the recovery likelihood because it is more tractable than marginal likelihood, as sampling from the conditional distributions is much easier than sampling from the marginal distributions.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    RePaint features an enhanced denoising strategy that uses resampling iterations to better condition the image.

    “For inpainting tasks, RePaint [146] features an enhanced denoising strategy that uses resampling iterations to better condition the image (see Figure 5).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    SDEdit uses a Stochastic Differential Equation prior to improve fidelity.

    “SDEdit [160] uses a Stochastic Differential Equation (SDE) prior to improve fidelity.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Palette employs conditional diffusion models to create a unified framework for four image generation tasks: colorization, inpainting, uncropping, and JPEG restoration.

    “Meanwhile, Palette [198] employs conditional diffusion models to create a unified framework for four image generation tasks: colorization, inpainting, uncropping, and JPEG restoration.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    SR3 conducts super-resolution through a stochastic, iterative denoising process.

    “SR3 conducts super-resolution through a stochastic, iterative denoising process.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    The Cascaded Diffusion Model consists of multiple diffusion models in sequence, each generating images of increasing resolution.

    “The Cascaded Diffusion Model (CDM) [88] consists of multiple diffusion models in sequence, each generating images of increasing resolution.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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  • May combine two claims.

    Premisecitation

    This model can generate high-quality samples, and long-run MCMC samples from the conditional distributions still resemble realistic images.

    “This model can generate high-quality samples, and long-run MCMC samples from the conditional distributions still resemble realistic images.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [69]

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  • May combine two claims.

    Premisereasoning

    Both SR3 and CDM directly apply the diffusion process to input images, which leads to larger evaluation steps.

    “Both the SR3 and CDM directly apply the diffusion process to input images, which leads to larger evaluation steps.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
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    Premisecitation

    Decoder Denoising Pretraining integrates diffusion models with denoising autoencoders and delivers promising results on label-efficient semantic segmentation.

    “Similarly, Decoder Denoising Pretraining (DDeP) [19] integrates diffusion models with denoising autoencoders [241] and delivers promising results on label-efficient semantic segmentation.”
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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    SR3 uses DDPM to enable conditional image generation.

    “Super-Resolution via Repeated Refinement (SR3) [200] uses DDPM to enable conditional image generation.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Diffusion models learn and even amplify the bias in the training dataset.

    “diffusion models would learn and even amplify the bias in the training dataset [21,39]”
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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Some approaches apply guided diffusion processes for advanced adversarial purification.

    “In addition, some approaches apply guided diffusion processes for advanced adversarial purification [242,251].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
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  • May combine two claims.

    Premisecitation

    Markov chains evolving with equivariant Markov kernels can produce an invariant distribution, and blocks for the Markov kernels can be designed to preserve the desirable equivariance property.

    “GeoDiff [261] demonstrates that Markov chains evolving with equivariant Markov kernels can produce an invariant distribution, and further designs blocks for the Markov kernels to preserve the desirable equivariance property”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Because the latent space in diffusion models often possesses the same dimensionality as the data space, sampling efficiency is negatively affected and the models may not learn the representation schemes well.

    “Furthermore, since the latent space in diffusion models often possesses the same dimensionality as the data space, sampling efficiency is negatively affected and the models may not learn the representation schemes well [103].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Wang et al. employ a diffusion model to improve adversarial training, and the trained models achieve state-of-the-art performance using only generated data.

    “Wang et al. [244] employ the recent diffusion model to improve adversarial training, and the trained models achieve state-ofthe-art performance using only generated data.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
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    Premisecitation

    Recent advances in Schrödinger bridges and optimal transport provide promising alternative solutions, suggesting new formulations for diffusion models that are capable of converging to a specified prior distribution in finite time.

    “Recent advances in Schrödinger bridges and optimal transport [35,50,52,214,221] provide promising alternative solutions, suggesting new formulations for diffusion models that are capable of converging to a specified prior distribution in finite time.”
    Quote in the paperNot a substantial claim0 of 5 sources checked
    • cites [35]

      Not checked in the free run.

    • cites [50]

      Not checked in the free run.

    • cites [52]

      Not checked in the free run.

    • cites [214]

      Not checked in the free run.

    • cites [221]

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Other methods incorporate the equivariance property into 3D molecule generation and protein generation.

    “Other methods incorporate the equivariance property into 3D molecule generation [91] and protein generation [3,15].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [91]

      Not checked in the free run.

    • cites [3]

      Not checked in the free run.

    • cites [15] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    Complete removal of information is unachievable in finite time.

    “In reality, complete removal of information is unachievable in finite time.”
    Quote in the paperNot a substantial claim
119 of 119

  • The paper relies on this without support.

    Over the past two years, the body of research on diffusion models has grown significantly, making it increasingly challenging for new researchers to stay abreast of recent developments.

    “Over the past two years, the body of research on diffusion models has grown significantly, making it increasingly challenging for new researchers to stay abreast of the recent developments in the field.”

  • The paper relies on this without support.

    The sheer volume of work on diffusion models can obscure major trends and hinder further research progress.

    “Additionally, the sheer volume of work can obscure major trends and hinder further research progress.”

  • The authors state this as an assumption.

    Langevin dynamics converges to its equilibrium at every noise level.

    “under the assumption that Langevin dynamics converges to its equilibrium at every noise level”

  • The paper relies on this without support.

    Text-to-image generation models struggle to faithfully reproduce complex sentences with many objects and relationships.

    “Despite text-to-image generation models have made significant progress, they struggle to faithfully reproduce complex sentences with many objects and relationships.”

  • The paper relies on this without support.

    High-quality paired scene-graph-image datasets are scarce.

    “However, highquality paired SG-image datasets are scarce.”

  • The paper relies on this without support.

    Diffusion models have emerged as a powerful framework, notably as the only one that can rival generative adversarial networks in most applications without resorting to adversarial training.

    “Diffusion models have emerged as a powerful framework, notably as the only one that can rival generative adversarial networks (GANs) in most applications without resorting to adversarial training.”

  • The paper relies on this without support.

    Unlike variational autoencoders or generative adversarial networks, diffusion models are less effective for providing good representations of data in their latent space.

    “Unlike variational autoencoders or generative adversarial networks, diffusion models are less effective for providing good representations of data in their latent space.”
7 of 7

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Diffusion models' ability to produce photorealistic imagery with fine-grained control has made them the dominant tools for generating synthetic content, including AI CSAM.

    “Their ability to produce photorealistic imagery with fine-grained control has made them the dominant tools for generating synthetic content (Yang et al. 2023), including AI CSAM (Thorn 2024a).”

    From AI-generated child sexual abuse material: what’s the harm? (2026) · cites this paper as Yang et al. 2023

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Grad-TTS2 name disagrees with its cited reference

    Minor

    The text calls the model Grad-TTS2, but reference [118] is titled Guided-TTS 2.

  • Latent Diffusion Models is cited as DPM-solver

    Minor

    The citation attached to Latent Diffusion Models points to a bibliography entry for DPM-solver, not the paper identified elsewhere in the text as LDM [196].

  • GeoDiff citation year disagrees with the reference

    Minor

    The text dates reference [261] to 2022, while its bibliography entry gives 2021.

  • DDPM reverse SDE has the opposite score-term sign

    Minor

    Equation (18) subtracts the score term in a reverse-time SDE. Substituting the DDPM drift from Equation (16) therefore gives a negative score term, but Equation (21) gives it a positive sign.

  • Annealed Langevin sampling uses incompatible time-step ranges

    Minor

    The stated iteration runs from T down to 1, but the immediately following initialization rule applies from 0 up to T−1. As written, the rule does not initialize the first stated step, t = T, and the stated iteration does not perform the t = 0 step.

5 of 5

Cites

Flagged first, then by citations

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Cited by

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Related

Authors
Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., & Yang, M.-H.
Affiliations
  • Peking University (CN)
  • OpenAI (United States) (US)
  • University of California, Los Angeles (US)
  • University of Southern California (US)
  • HEC Montréal (CA)
  • University of California, Merced (US)
ORCID
Venue
ACM Computing Surveys
Publisher
Association for Computing Machinery (ACM)
Published
2023-11-09
Volume
56
Issue
4
Pages
1-39
ISSN
  • 0360-0300 (ISSN-L)
  • 1557-7341
DOI
10.1145/3626235
OpenAlex
W4387195417
Cited by
1,704
References
199
Citations by year
2026: 392 · 2025: 683 · 2024: 483 · 2023: 142 · 2022: 4
FWCI
154.02
Type
journal article
Language
en
Open access
Licence
www.acm.org/publications/policies/copyright_policy#Background (vor)
Locations
  • ACM Computing Surveys (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Generative Adversarial Networks and Image Synthesis
  • Physical Sciences › Computer Science › Artificial Intelligence › Adversarial Robustness in Machine Learning
  • Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image and Signal Denoising Methods
Keywords
diffusion models, deep generative models, efficient sampling, likelihood estimation, structured data, hybrid generative models, image generation, video generation, temporal data modeling

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-10.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-10.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

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