mlsanityData to 14 Oct 2026

Convergence of denoising diffusion models under the manifold hypothesis

arXiv 2208.05314 ↗First posted 10 Aug 2022stat.ML cs.LG math.PRcited by 242

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Denoising diffusion models are a recent class of generative models exhibiting state-of-the-art performance in image and audio synthesis. Such models approximate the time-reversal of a forward noising process from a target distribution to a reference density, which is usually Gaussian. Despite their strong empirical results, the theoretical analysis of such models remains limited. In particular, all current approaches crucially assume that the target density admits a density w.r.t. the Lebesgue measure. This does not cover settings where the target distribution is supported on a lower-dimensional manifold or is given by some empirical distribution. In this paper, we bridge this gap by providing the first convergence results for diffusion models in this more general setting. In particular, we provide quantitative bounds on the Wasserstein distance of order one between the target data distribution and the generative distribution of the diffusion model.

Key facts. Cited by 242 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 26 of its 66 references that are in the map. First posted on arXiv 10 Aug 2022.

Builds on

26 of 66 references.

Closest ancestors · 17

Same direction, newest first

  1. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  2. 2022How Much is Enough? A Study on Diffusion Times in Score-based Generative Models10 Jun
  3. 2022Score-Based Generative Models Detect Manifolds2 Jun
  4. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  5. 2022Riemannian Score-Based Generative Modelling6 Feb
  6. 2021Tackling the Generative Learning Trilemma with Denoising Diffusion GANs15 Dec
  7. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusion14 Dec
  8. 2021Score-based Generative Modeling in Latent Space10 Jun
  9. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation10 Jun
  10. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  11. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  12. 2021Diffusion Models Beat GANs on Image Synthesis11 May
5 more
  1. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  2. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  4. 2020Denoising Diffusion Probabilistic Models19 Jun
  5. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 9

  1. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  2. 2017Learning Deep CNN Denoiser Prior for Image Restorationcited by 408
  3. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 208
  4. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 197
  5. 2021How Faithful is your Synthetic Data? Sample-level Metrics for Evaluating and Auditing Generative Modelscited by 196
  6. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  7. 2020Adversarial score matching and improved sampling for image generationcited by 96
  8. 2022Diffusion bridges vector quantized Variational AutoEncoderscited by 16
  9. 2020Fast Mixing of Multi-Scale Langevin Dynamics under the Manifold Hypothesiscited by 11

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 8; 2023: 37; 2024: 78; 2025: 60; 2026: 59.

242 papers.

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  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 445
  2. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 235
  3. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 219
  4. 2022Convergence of score-based generative modeling for general data distributionscited by 181
  5. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 174
  6. 2023Denoising Diffusion Samplerscited by 149
  7. 2022An optimal control perspective on diffusion-based generative modelingcited by 147
  8. 2023The probability flow ODE is provably fastcited by 146
  9. 2024Dynamical Regimes of Diffusion Modelscited by 142
  10. 2024An analytic theory of creativity in convolutional diffusion modelscited by 104
  11. 2022Refining Generative Process with Discriminator Guidance in Score-based Diffusion Modelscited by 100
  12. 2024A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Datacited by 93
38 more
  1. 2023Error Bounds for Flow Matching Methodscited by 91
  2. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Modelscited by 85
  3. 2023DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuningcited by 84
  4. 2025Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Trainingcited by 81
  5. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 77
  6. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 76
  7. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplerscited by 74
  8. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 73
  9. 2024Accelerating Diffusion Sampling with Optimized Time Stepscited by 67
  10. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 65
  11. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptionscited by 63
  12. 2024Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theorycited by 63
  13. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  14. 2024Gradient Guidance for Diffusion Models: An Optimization Perspectivecited by 61
  15. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 60
  16. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  17. 2022From Denoising Diffusions to Denoising Markov Modelscited by 58
  18. 2023Learning Mixtures of Gaussians Using the DDPM Objectivecited by 57
  19. 2023Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimizationcited by 57
  20. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  21. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Modelscited by 56
  22. 2024On the Trajectory Regularity of ODE-based Diffusion Samplingcited by 55
  23. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributionscited by 53
  24. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 50
  25. 2024Improved off-policy training of diffusion samplerscited by 48
  26. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptionscited by 47
  27. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion modelscited by 47
  28. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  29. 2023Closed-Form Diffusion Modelscited by 46
  30. 2024Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Modelscited by 46
  31. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transformcited by 46
  32. 2024Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformizationcited by 44
  33. 2024Learning Mixtures of Gaussians Using Diffusion Modelscited by 42
  34. 2024A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Modelscited by 42
  35. 2023Infinite-Dimensional Diffusion Modelscited by 39
  36. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 38
  37. 2023Analysis of learning a flow-based generative model from limited sample complexitycited by 38
  38. 2024Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structurecited by 38

50 most cited of 242 All 242 (JSON)

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  1. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  2. 2023Diffusion Models for Constrained Domains
  3. 2025Low-dimensional adaptation of diffusion models: Convergence in total variation
  4. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesis
  5. 2024Statistical guarantees for denoising reflected diffusion models
  6. 2026PTL-Diffusion: Manifold-Aware Diffusion with Periodic Terminal Laws
  7. 2025Diffusion Models: A Mathematical Introduction
  8. 2021Variational Diffusion Models