mlsanityData to 7 Oct 2026

Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

arXiv 2409.18804 ↗First posted 27 Sep 2024stat.ML cs.LG math.STcited by 60

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more applications in science and beyond. The textitmanifold hypothesis states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of score learning. In terms of sampling complexity, we obtain rates independent of the ambient dimension w.r.t. the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.

Key facts. Cited by 60 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 1 of its 48 references that are in the map. First posted on arXiv 27 Sep 2024.

Builds on

1 of 48 references.

References · 1

  1. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 26; 2026: 29.

60 papers.

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  1. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  2. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesiscited by 42
  3. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 38
  4. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  5. 2025Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distancecited by 20
  6. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  7. 2026Confidence-Based Decoding is Provably Efficient for Diffusion Language Modelscited by 17
  8. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
  9. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
  10. 2025A solvable model of learning generative diffusion: theory and insightscited by 14
  11. 2025When Scores Learn Geometry: Rate Separations under the Manifold Hypothesiscited by 13
  12. 2026High-accuracy sampling for diffusion models and log-concave distributionscited by 13
38 more
  1. 2024Shallow diffusion networks provably learn hidden low-dimensional structurecited by 12
  2. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 9
  3. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  4. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  5. 2025Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spacescited by 8
  6. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Boundscited by 8
  7. 2026Flow Matching is Adaptive to Manifold Structurescited by 8
  8. 2025Nonparametric estimation of a factorizable density using diffusion modelscited by 7
  9. 2025Diffusion and Flow-based Copulas: Forgetting and Remembering Dependenciescited by 7
  10. 2025Towards a unified framework for guided diffusion modelscited by 7
  11. 2024Statistical guarantees for denoising reflected diffusion modelscited by 6
  12. 2025Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysiscited by 6
  13. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexitycited by 6
  14. 2025Localized Diffusion Modelscited by 5
  15. 2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 5
  16. 2025Optimal Denoising in Score-Based Generative Models: The Role of Data Regularitycited by 4
  17. 2025Algorithm- and Data-Dependent Generalization Bounds for Diffusion Modelscited by 4
  18. 2026Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guaranteescited by 4
  19. 2026Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Datacited by 4
  20. 2025Dimension-free error estimate for diffusion model and optimal schedulingcited by 3
  21. 2025Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networkscited by 3
  22. 2026Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Datacited by 3
  23. 2026Benign Overfitting Does Not Occur in Diffusion Modelscited by 3
  24. 2025Approximation Capabilities of Feedforward Neural Networks with GELU Activationscited by 2
  25. 2026Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspectivecited by 2
  26. 2026Manifold Generalization Provably Proceeds Memorization in Diffusion Modelscited by 2
  27. 2026Understanding diffusion models requires rethinking (again) generalizationcited by 2
  28. 2026Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributionscited by 2
  29. 2026Denoising growth complexity: Data geometry and certified schedules for diffusion samplingcited by 2
  30. 2025Expressive Power of Implicit Models: Rich Equilibria and Test-Time Scalingcited by 1
  31. 2025Optimal Stopping in Latent Diffusion Modelscited by 1
  32. 2026Reflected diffusion models adapt to low-dimensional datacited by 1
  33. 2026On the Limits of Latent Reuse in Diffusion Modelscited by 1
  34. 2026Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choicescited by 1
  35. 2025Model-free filtering in high dimensions via projection and score-based diffusionscited by 0
  36. 2026Transformers Learn the Optimal DDPM Denoiser for Multi-Token GMMscited by 0
  37. 2026Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problemscited by 0
  38. 2026Statistical Convergence of Spherical First Hitting Diffusion Modelscited by 0

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  1. 2022Convergence of denoising diffusion models under the manifold hypothesis
  2. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models
  3. 2025MAD: Manifold Attracted Diffusion
  4. 2022Your diffusion model secretly knows the dimension of the data manifold
  5. 2022Pseudo Numerical Methods for Diffusion Models on Manifolds
  6. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity
  7. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models
  8. 2023Diffusion Models for Constrained Domains