mlsanityData to 7 Oct 2026

Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions

arXiv 2209.11215 ↗First posted 22 Sep 2022cs.LG math.ST stat.THcited by 440

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL·E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to prior works, our results (1) hold for an L2-accurate score estimate (rather than L^∞-accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small. We view this as strong theoretical justification for the empirical success of SGMs. We also examine SGMs based on the critically damped Langevin diffusion (CLD). Contrary to conventional wisdom, we provide evidence that the use of the CLD does not reduce the complexity of SGMs.

Key facts. Cited by 440 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 33 of its 54 references that are in the map. First posted on arXiv 22 Sep 2022.

Builds on

33 of 54 references.

Closest ancestors · 15

Same direction, newest first

  1. 2022Diffusion Models in Vision: A Survey10 Sep
  2. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  3. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models31 Aug
  4. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  5. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  6. 2022Score-Based Generative Models Detect Manifolds2 Jun
  7. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  8. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusion14 Dec
  9. 2021Variational Diffusion Models1 Jul
  10. 2021Score-based Generative Modeling in Latent Space10 Jun
  11. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  12. 2021Diffusion Models Beat GANs on Image Synthesis11 May
3 more
  1. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  3. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 18

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 224
  5. 2017Underdamped Langevin MCMC: A non-asymptotic analysiscited by 131
  6. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 118
  7. 2016Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixturescited by 114
  8. 2020Structured Logconcave Sampling with a Restricted Gaussian Oraclecited by 85
  9. 2019The Randomized Midpoint Method for Log-Concave Samplingcited by 77
  10. 2022Improved analysis for a proximal algorithm for samplingcited by 72
  11. 2018On sampling from a log-concave density using kinetic Langevin diffusionscited by 61
  12. 2020Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithmcited by 57
6 more
  1. 2020Continuous LWEcited by 37
  2. 2022Hardness of Noise-Free Learning for Two-Hidden-Layer Neural Networkscited by 27
  3. 2022Continuous LWE is as Hard as LWE & Applications to Learning Gaussian Mixturescited by 26
  4. 2021From Local Pseudorandom Generators to Hardness of Learningcited by 24
  5. 2019Bounding the error of discretized Langevin algorithms for non-strongly log-concave targetscited by 16
  6. 2022Learning (Very) Simple Generative Models Is Hardcited by 9

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 6; 2023: 67; 2024: 128; 2025: 130; 2026: 109.

440 papers.

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  1. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  2. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  3. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusioncited by 448
  4. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  5. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 215
  6. 2022Convergence of score-based generative modeling for general data distributionscited by 179
  7. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 171
  8. 2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 159
  9. 2023Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Modelscited by 156
  10. 2023DDP: Diffusion Model for Dense Visual Predictioncited by 148
  11. 2022An optimal control perspective on diffusion-based generative modelingcited by 145
  12. 2023The probability flow ODE is provably fastcited by 145
38 more
  1. 2023Denoising Diffusion Samplerscited by 144
  2. 2024Dynamical Regimes of Diffusion Modelscited by 133
  3. 2023Ambient Diffusion: Learning Clean Distributions from Corrupted Datacited by 108
  4. 2023Policy Representation via Diffusion Probability Model for Reinforcement Learningcited by 105
  5. 2025Masked Autoencoders Are Effective Tokenizers for Diffusion Modelscited by 93
  6. 2024Align Your Steps: Optimizing Sampling Schedules in Diffusion Modelscited by 91
  7. 2023Error Bounds for Flow Matching Methodscited by 88
  8. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Modelscited by 84
  9. 2023DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuningcited by 82
  10. 2024Classifier-Free Guidance is a Predictor-Correctorcited by 79
  11. 2023Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusioncited by 78
  12. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 74
  13. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplerscited by 73
  14. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
  15. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 72
  16. 2023On the Generalization Properties of Diffusion Modelscited by 67
  17. 2024Accelerating Diffusion Sampling with Optimized Time Stepscited by 67
  18. 2023Particle Guidance: non-I.I.D. Diverse Sampling with Diffusion Modelscited by 65
  19. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 64
  20. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 63
  21. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptionscited by 61
  22. 2023The Blessing of Randomness: SDE Beats ODE in General Diffusion-based Image Editingcited by 61
  23. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  24. 2024Optimal score estimation via empirical Bayes smoothingcited by 60
  25. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 59
  26. 2024Gradient Guidance for Diffusion Models: An Optimization Perspectivecited by 59
  27. 2024DPLM-2: A Multimodal Diffusion Protein Language Modelcited by 59
  28. 2022From Denoising Diffusions to Denoising Markov Modelscited by 58
  29. 2023Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistentcited by 58
  30. 2024Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theorycited by 58
  31. 2023Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimizationcited by 57
  32. 2023Learning Mixtures of Gaussians Using the DDPM Objectivecited by 56
  33. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  34. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Modelscited by 56
  35. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Modelscited by 56
  36. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributionscited by 53
  37. 2023Reward-Directed Conditional Diffusion: Provable Distribution Estimation and Reward Improvementcited by 49
  38. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 48

50 most cited of 440 All 440 (JSON)

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  3. 2023Closed-Form Diffusion Models
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  5. 2023Preconditioned Score-based Generative Models
  6. 2026From Score Approximation to Distribution Approximation in Score-Based Diffusion Models
  7. 2023A Complete Recipe for Diffusion Generative Models
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