mlsanityData to 14 Oct 2026

Diffusion Models are Minimax Optimal Distribution Estimators

arXiv 2303.01861 ↗First posted 3 Mar 2023stat.ML cs.LGcited by 165

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling for well-known function spaces. The highlight of this paper is that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates in the total variation distance and in the Wasserstein distance of order one. Furthermore, we extend our theory to demonstrate how diffusion models adapt to low-dimensional data distributions. We expect these results advance theoretical understandings of diffusion modeling and its ability to generate verisimilar outputs.

Key facts. Cited by 165 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 8 of its 63 references that are in the map. First posted on arXiv 3 Mar 2023.

Builds on

8 of 63 references.

Closest ancestors · 2

Same direction, newest first

  1. 2022Your diffusion model secretly knows the dimension of the data manifold23 Dec
  2. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep

Other references · 6

  1. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  2. 2022Video Diffusion Modelscited by 2,305
  3. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  4. 2019Deep ReLU network approximation of functions on a manifoldcited by 74
  5. 2017How Well Can Generative Adversarial Networks Learn Densities: A Nonparametric Viewcited by 14
  6. 2018Minimax Distribution Estimation in Wasserstein Distancecited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 15; 2024: 53; 2025: 48; 2026: 49.

165 papers.

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  1. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 235
  2. 2024An analytic theory of creativity in convolutional diffusion modelscited by 104
  3. 2023On the Stability of Iterative Retraining of Generative Models on their own Datacited by 100
  4. 2024A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Datacited by 93
  5. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Modelscited by 85
  6. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 73
  7. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 65
  8. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptionscited by 63
  9. 2024Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theorycited by 63
  10. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  11. 2024Gradient Guidance for Diffusion Models: An Optimization Perspectivecited by 61
  12. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 60
38 more
  1. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Modelscited by 56
  2. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributionscited by 53
  3. 2023Reward-Directed Conditional Diffusion: Provable Distribution Estimation and Reward Improvementcited by 49
  4. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptionscited by 47
  5. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  6. 2023Closed-Form Diffusion Modelscited by 46
  7. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesiscited by 45
  8. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalizationcited by 42
  9. 2024Learning Mixtures of Gaussians Using Diffusion Modelscited by 42
  10. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Modelscited by 39
  11. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 39
  12. 2023Analysis of learning a flow-based generative model from limited sample complexitycited by 38
  13. 2024Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structurecited by 38
  14. 2024A Good Score Does not Lead to A Good Generative Modelcited by 36
  15. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
  16. 2024Contractive Diffusion Probabilistic Modelscited by 29
  17. 2024Flow matching achieves almost minimax optimal convergencecited by 29
  18. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distancecited by 29
  19. 2024From optimal score matching to optimal samplingcited by 28
  20. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  21. 2024Evaluating the design space of diffusion-based generative modelscited by 26
  22. 2023On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimatescited by 25
  23. 2025How Compositional Generalization and Creativity Improve as Diffusion Models are Trainedcited by 25
  24. 2025Bigger Isn't Always Memorizing: Early Stopping Overparameterized Diffusion Modelscited by 25
  25. 2024Diffusion Model for Data-Driven Black-Box Optimizationcited by 24
  26. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  27. 2025Breaking AR's Sampling Bottleneck: Provable Acceleration via Diffusion Language Modelscited by 23
  28. 2023Distribution learning via neural differential equations: a nonparametric statistical perspectivecited by 22
  29. 2025Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distancecited by 21
  30. 2024A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Modelscited by 20
  31. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  32. 2026Confidence-Based Decoding is Provably Efficient for Diffusion Language Modelscited by 18
  33. 2024U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Modelscited by 17
  34. 2024DiffPO: A causal diffusion model for learning distributions of potential outcomescited by 17
  35. 2024Convergence Analysis of Flow Matching in Latent Space with Transformerscited by 16
  36. 2024Diffusing States and Matching Scores: A New Framework for Imitation Learningcited by 16
  37. 2024Score-based generative models are provably robust: an uncertainty quantification perspectivecited by 15
  38. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15

50 most cited of 165 All 165 (JSON)

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  1. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data
  2. 2026Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data
  3. 2023A Note on the Convergence of Denoising Diffusion Probabilistic Models
  4. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Models
  5. 2023On the Generalization of Diffusion Model
  6. 2023The Emergence of Reproducibility and Generalizability in Diffusion Models
  7. 2023On the Generalization Properties of Diffusion Models
  8. 2025Diagnosing and Improving Diffusion Models by Estimating the Optimal Loss Value