mlsanityData to 14 Oct 2026

Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances

arXiv 2401.17958 ↗First posted 31 Jan 2024stat.ML cs.LG math.PRcited by 56

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative modeling with probability flow ordinary differential equations (ODEs) has achieved remarkable success in a variety of applications. While various fast ODE-based samplers have been proposed in the literature and employed in practice, the theoretical understandings about convergence properties of the probability flow ODE are still quite limited. In this paper, we provide the first non-asymptotic convergence analysis for a general class of probability flow ODE samplers in 2-Wasserstein distance, assuming accurate score estimates and smooth log-concave data distributions. We then consider various examples and establish results on the iteration complexity of the corresponding ODE-based samplers. Our proof technique relies on spelling out explicitly the contraction rate for the continuous-time ODE and analyzing the discretization and score-matching errors using synchronous coupling; the challenge in our analysis mainly arises from the inherent non-autonomy of the probability flow ODE and the specific exponential integrator that we study.

Key facts. Cited by 56 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 30 of its 46 references that are in the map. First posted on arXiv 31 Jan 2024.

Builds on

30 of 46 references.

Closest ancestors · 18

Same direction, newest first

  1. 2023On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates22 Nov
  2. 2023Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models18 Nov
  3. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  4. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models15 Jun
  5. 2023The probability flow ODE is provably fast19 May
  6. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  7. 2023Consistency Models2 Mar
  8. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  9. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  10. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  11. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  12. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
6 more
  1. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  2. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  3. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  4. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  5. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  6. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov

Other references · 12

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  3. 2022Flow Matching for Generative Modelingcited by 5,702
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
  6. 2021Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speechcited by 447
  7. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  8. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  9. 2017User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradientcited by 116
  10. 2023The Blessing of Randomness: SDE Beats ODE in General Diffusion-based Image Editingcited by 62
  11. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  12. 2020Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlocited by 19

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 21; 2025: 21; 2026: 14.

56 papers.

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  1. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 76
  2. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 65
  3. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  4. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Modelscited by 57
  5. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 50
  6. 2024How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Frameworkcited by 48
  7. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  8. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  9. 2024Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexitycited by 42
  10. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 39
  11. 2025Stochastic Self-Guidance for Training-Free Enhancement of Diffusion Modelscited by 36
  12. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
38 more
  1. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  2. 2024Convergence of Continuous Normalizing Flows for Learning Probability Distributionscited by 26
  3. 2024Evaluating the design space of diffusion-based generative modelscited by 26
  4. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  5. 2025Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distancecited by 21
  6. 2024Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approachcited by 19
  7. 2025Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Accelerationcited by 17
  8. 2024Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errorscited by 16
  9. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
  10. 2025Optimal Convergence Analysis of DDPM for General Distributionscited by 16
  11. 2026High-accuracy sampling for diffusion models and log-concave distributionscited by 16
  12. 2024Theoretical guarantees in KL for Diffusion Flow Matchingcited by 14
  13. 2024Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplerscited by 14
  14. 2024Shallow diffusion networks provably learn hidden low-dimensional structurecited by 12
  15. 2024Wasserstein Bounds for generative diffusion models with Gaussian tail targetscited by 10
  16. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Boundscited by 9
  17. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  18. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  19. 2025A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptionscited by 8
  20. 2026Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalitiescited by 8
  21. 2024Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplerscited by 7
  22. 2025Non-asymptotic error bounds for probability flow ODEs under weak log-concavitycited by 7
  23. 2025Sublinear iterations can suffice even for DDPMscited by 7
  24. 2024Generating Physical Dynamics under Priorscited by 6
  25. 2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 6
  26. 2025Parallel Sampling via Autospeculationcited by 6
  27. 2025Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based modelscited by 4
  28. 2025Elucidating Rectified Flow with Deterministic Sampler: Polynomial Discretization Complexity for Multi and One-step Modelscited by 3
  29. 2025Diffusion Generative Models Meet Compressed Sensing, with Applications to Imaging and Financecited by 3
  30. 2025Dimension-free error estimate for diffusion model and optimal schedulingcited by 3
  31. 2025Multi-Step Consistency Models: Fast Generation with Theoretical Guaranteescited by 2
  32. 2026Query Lower Bounds for Diffusion Samplingcited by 2
  33. 2025Non-asymptotic convergence bound of conditional diffusion modelscited by 1
  34. 2026Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplerscited by 1
  35. 2026Noise Schedule Design for Diffusion Models: An Optimal Control Perspectivecited by 1
  36. 2026Diffusion Models for Adaptive Sequential Data Generationcited by 1
  37. 2026Wasserstein Convergence of ODE-Based Samplers in Decentralized Diffusion Model via Velocity Field Decompositioncited by 1
  38. 2026Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspectivecited by 0

50 most cited of 56 All 56 (JSON)

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  1. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Models
  2. 2026Probability flow ODEs in score-based and reflected diffusion models
  3. 2026Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees
  4. 2026Formalizing the Sampling Design Space of Diffusion-Based Generative Models via Adaptive Solvers and Wasserstein-Bounded Timesteps
  5. 2023Error Bounds for Flow Matching Methods
  6. 2023On Accelerating Diffusion-Based Sampling Process via Improved Integration Approximation
  7. 2024Flow matching achieves almost minimax optimal convergence
  8. 2024Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences