mlsanityData to 14 Oct 2026

Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions

arXiv 2211.01916 ↗First posted 3 Nov 2022cs.LGcited by 219

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in ε-accuracy can be done in tilde Oleft(fracd log (1/δ)εright) steps: 1) the variance-δ Gaussian perturbation of any data distribution; 2) data distributions with 1/δ-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

Key facts. Cited by 219 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 28 of its 40 references that are in the map. First posted on arXiv 3 Nov 2022.

Builds on

28 of 40 references.

Closest ancestors · 14

Same direction, newest first

  1. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  2. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  3. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  4. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  5. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  6. 2022Score-Based Generative Models for Molecule Generation7 Mar
  7. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  8. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusion14 Dec
  9. 2021Structured Denoising Diffusion Models in Discrete State-Spaces7 Jul
  10. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  11. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  12. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
2 more
  1. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  2. 2020Denoising Diffusion Implicit Models6 Oct

Other references · 14

  1. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  2. 2015Variational Inference with Normalizing Flowscited by 2,170
  3. 2022Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learningcited by 562
  4. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 521
  5. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  6. 2021Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contractioncited by 307
  7. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 227
  8. 2021Learning Gradient Fields for Molecular Conformation Generationcited by 163
  9. 2016Energy-based Generative Adversarial Networkcited by 119
  10. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  11. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64
  12. 2022Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localizationcited by 48
2 more
  1. 2022Probability flow solution of the Fokker-Planck equationcited by 47
  2. 2021Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flowscited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 29; 2024: 64; 2025: 70; 2026: 56.

219 papers.

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  1. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
  2. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 737
  3. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 235
  4. 2023The probability flow ODE is provably fastcited by 146
  5. 2023Policy Representation via Diffusion Probability Model for Reinforcement Learningcited by 112
  6. 2025Masked Autoencoders Are Effective Tokenizers for Diffusion Modelscited by 96
  7. 2023Error Bounds for Flow Matching Methodscited by 91
  8. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Modelscited by 85
  9. 2023White-Box Transformers via Sparse Rate Reductioncited by 80
  10. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 76
  11. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplerscited by 74
  12. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
38 more
  1. 2023On the Generalization Properties of Diffusion Modelscited by 68
  2. 2024Accelerating Diffusion Sampling with Optimized Time Stepscited by 67
  3. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 65
  4. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptionscited by 63
  5. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  6. 2023Learning Mixtures of Gaussians Using the DDPM Objectivecited by 57
  7. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Modelscited by 57
  8. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  9. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Modelscited by 56
  10. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributionscited by 53
  11. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 50
  12. 2024What does guidance do? A fine-grained analysis in a simple settingcited by 48
  13. 2024How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Frameworkcited by 48
  14. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptionscited by 47
  15. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion modelscited by 47
  16. 2024Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Modelscited by 46
  17. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesiscited by 45
  18. 2024Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformizationcited by 44
  19. 2024Learning general Gaussian mixtures with efficient score matchingcited by 44
  20. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  21. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalizationcited by 42
  22. 2024Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexitycited by 42
  23. 2023Reverse Diffusion Monte Carlocited by 41
  24. 2023Infinite-Dimensional Diffusion Modelscited by 39
  25. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Modelscited by 39
  26. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 39
  27. 2023Analysis of learning a flow-based generative model from limited sample complexitycited by 38
  28. 2023A Theoretical Justification for Image Inpainting using Denoising Diffusion Probabilistic Modelscited by 37
  29. 2024A Good Score Does not Lead to A Good Generative Modelcited by 36
  30. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
  31. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  32. 2024Flow matching achieves almost minimax optimal convergencecited by 29
  33. 2024Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallelcited by 29
  34. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distancecited by 29
  35. 2024An analysis of the noise schedule for score-based generative modelscited by 27
  36. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  37. 2024Towards a mathematical theory for consistency training in diffusion modelscited by 26
  38. 2024Evaluating the design space of diffusion-based generative modelscited by 26

50 most cited of 219 All 219 (JSON)

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  1. 2023Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models
  2. 2024Discrete vs. Continuous Trade-offs for Generative Models
  3. 2022Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance
  4. 2023Convergence guarantee for consistency models
  5. 2022Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models
  6. 2024Score-based Generative Models with Adaptive Momentum
  7. 2026From Global to Tail Convergence: Dynamics of α-Stable Score-Based Generative Models
  8. 2023Preconditioned Score-based Generative Models