mlsanityData to 14 Oct 2026

Distribution estimation via Flow Matching with Lipschitz guarantees

arXiv 2509.02337 ↗First posted 2 Sep 2025stat.ML cs.LG math.STcited by 6

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

Flow Matching, a promising approach in generative modeling, has recently gained popularity. Relying on ordinary differential equations, it offers a simple and flexible alternative to diffusion models, which are currently the state-of-the-art. Despite its empirical success, the mathematical understanding of its statistical power so far is very limited. This is largely due to the sensitivity of theoretical bounds to the Lipschitz constant of the vector field which drives the ODE. In this work, we study the assumptions that lead to controlling this dependency. Based on these results, we derive a convergence rate for the Wasserstein 1 distance between the estimated distribution and the target distribution which improves previous results in high dimensional setting. This rate applies to certain classes of unbounded distributions and particularly does not require log-concavity.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 47 of its 74 references that are in the map. First posted on arXiv 2 Sep 2025.

Builds on

47 of 74 references.

Closest ancestors · 13

Same direction, newest first

  1. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimation17 Apr
  2. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows12 Mar
  3. 2024Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold26 Aug
  4. 2024Discrete Flow Matching22 Jul
  5. 2024Text-to-Image Rectified Flow as Plug-and-Play Priors5 Jun
  6. 2024Flow matching achieves almost minimax optimal convergence31 May
  7. 2024Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation30 Apr
  8. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generation3 Oct
  9. 2023Functional Flow Matching26 May
  10. 2023Flow Matching on General Geometries7 Feb
  11. 2023Improving and generalizing flow-based generative models with minibatch optimal transport1 Feb
  12. 2022Flow Matching for Generative Modeling6 Oct
1 more
  1. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 34

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  3. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesiscited by 4,181
  4. 2018Neural Ordinary Differential Equationscited by 3,779
  5. 2015Variational Inference with Normalizing Flowscited by 2,170
  6. 2014NICE: Non-linear Independent Components Estimationcited by 1,058
  7. 2022Building Normalizing Flows with Stochastic Interpolantscited by 995
  8. 2016Error bounds for approximations with deep ReLU networkscited by 574
  9. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 445
  10. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 433
  11. 2022A Survey on Generative Diffusion Modelcited by 240
  12. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 219
22 more
  1. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 174
  2. 2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 165
  3. 2018Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionalitycited by 162
  4. 2019A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Normcited by 106
  5. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  6. 2023Error Bounds for Flow Matching Methodscited by 91
  7. 2018How Well Generative Adversarial Networks Learn Distributionscited by 78
  8. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 65
  9. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensionscited by 65
  10. 2014Control functionals for Monte Carlo integrationcited by 59
  11. 2023Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Modelscited by 58
  12. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  13. 2023VoiceFlow: Efficient Text-to-Speech with Rectified Flow Matchingcited by 49
  14. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptionscited by 47
  15. 2020Statistical guarantees for generative models without dominationcited by 27
  16. 2024Convergence of Continuous Normalizing Flows for Learning Probability Distributionscited by 26
  17. 2023On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimatescited by 25
  18. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  19. 2025Generalization error bound for denoising score matching under relaxed manifold assumptioncited by 20
  20. 2023Gaussian Interpolation Flowscited by 17
  21. 2025Optimal Scheduling of Dynamic Transportcited by 13
  22. 2024A Wasserstein perspective of Vanilla GANscited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 2; 2026: 4.

6 papers.

  1. 2026Flow Matching is Adaptive to Manifold Structurescited by 10
  2. 2026Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalitiescited by 8
  3. 2026A Kinetic Energy Perspective of Flow Matchingcited by 3
  4. 2025On The Hidden Biases of Flow Matching Samplerscited by 2
  5. 2025Adaptive Path Integral Diffusion: AdaPIDcited by 1
  6. 2026When do explicit penalties improve flow matching?cited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees
  2. 2026Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions
  3. 2024Theoretical guarantees in KL for Diffusion Flow Matching
  4. 2025On Flow Matching KL Divergence
  5. 2025Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching
  6. 2026Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data
  7. 2024Convergence Analysis of Flow Matching in Latent Space with Transformers
  8. 2025Non-asymptotic error bounds for probability flow ODEs under weak log-concavity