mlsanityData to 7 Oct 2026

Linear Convergence of Diffusion Models Under the Manifold Hypothesis

arXiv 2410.09046 ↗First posted 11 Oct 2024stat.ML cs.LG math.STcited by 42

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower d-dimensional manifold embedded into D-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in D or polynomial (superlinear) in d. The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension d. Moreover, we show that this linear dependency is sharp.

Key facts. Cited by 42 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 16 of its 27 references that are in the map. First posted on arXiv 11 Oct 2024.

Builds on

16 of 27 references.

Closest ancestors · 12

Same direction, newest first

  1. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions27 Sep
  2. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models23 May
  3. 2024Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models3 Mar
  4. 2023Generalization in diffusion models arises from geometry-adaptive harmonic representations4 Oct
  5. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  6. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  7. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  8. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  9. 2022Score-Based Generative Models Detect Manifolds2 Jun
  10. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  11. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  12. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 4

  1. 2023Discrete Diffusion Modeling by Estimating the Ratios of the Data Distributioncited by 712
  2. 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 307
  3. 2024Fast Timing-Conditioned Latent Audio Diffusioncited by 178
  4. 2024Optimal score estimation via empirical Bayes smoothingcited by 60

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 19; 2026: 21.

42 papers.

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  1. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  2. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 38
  3. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  4. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  5. 2024On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimalitycited by 21
  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. 2026High-accuracy sampling for diffusion models and log-concave distributionscited by 13
  11. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  12. 2026Adaptation to Intrinsic Dependence in Diffusion Language Modelscited by 9
30 more
  1. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  2. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Boundscited by 8
  3. 2026Blind denoising diffusion models and the blessings of dimensionalitycited by 8
  4. 2025Sublinear iterations can suffice even for DDPMscited by 7
  5. 2025Towards a unified framework for guided diffusion modelscited by 7
  6. 2026High-accuracy and dimension-free sampling with diffusionscited by 7
  7. 2025Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysiscited by 6
  8. 2025Localized Diffusion Modelscited by 5
  9. 2026Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Datacited by 4
  10. 2025Diffusion Generative Models Meet Compressed Sensing, with Applications to Imaging and Financecited by 3
  11. 2025Dimension-free error estimate for diffusion model and optimal schedulingcited by 3
  12. 2026Polynomial Convergence of Riemannian Diffusion Modelscited by 2
  13. 2026Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matchingcited by 2
  14. 2026Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspectivecited by 2
  15. 2026Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributionscited by 2
  16. 2026Denoising growth complexity: Data geometry and certified schedules for diffusion samplingcited by 2
  17. 2025Expressive Power of Implicit Models: Rich Equilibria and Test-Time Scalingcited by 1
  18. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Modelscited by 1
  19. 2026Entropy-Based Dimension-Free Convergence and Loss-Adaptive Schedules for Diffusion Modelscited by 1
  20. 2026Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choicescited by 1
  21. 2025Concentration Inequalities for Stochastic Optimization of Unbounded Objective Functions with Application to Denoising Score Matchingcited by 0
  22. 2025MAD: Manifold Attracted Diffusioncited by 0
  23. 2026Causality-Encoded Diffusion Models for Interventional Sampling and Edge Inferencecited by 0
  24. 2026When Diffusion Model Can Ignore Dimension: An Entropy-Based Theorycited by 0
  25. 2026Sampling from Flow Language Models via Marginal-Conditioned Bridgescited by 0
  26. 2026Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refinecited by 0
  27. 2026Why Are DMD Students Lazy? Understanding the Copying Behavior in Few-Step Distillationcited by 0
  28. 2026Diffusion Models for High-Dimensional Clustered Data: Intrinsic-Dimension Adaptivity via Bayesian Classificationcited by 0
  29. 2026A statistical theory for blind denoising: minimax estimation of the noise level from a single samplecited by 0
  30. 2026How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectracited by 0

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  1. 2022Convergence of denoising diffusion models under the manifold hypothesis
  2. 2025Analysis of Diffusion Models for Manifold Data
  3. 2023Infinite-Dimensional Diffusion Models
  4. 2026Manifold Generalization Provably Proceeds Memorization in Diffusion Models
  5. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  6. 2026Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature
  7. 2024Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion
  8. 2025A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions