mlsanityData to 7 Oct 2026

Flow matching achieves almost minimax optimal convergence

arXiv 2405.20879 ↗First posted 31 May 2024cs.LGcited by 27

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

Abstract (arXiv)

Flow matching (FM) has gained significant attention as a simulation-free generative model. Unlike diffusion models, which are based on stochastic differential equations, FM employs a simpler approach by solving an ordinary differential equation with an initial condition from a normal distribution, thus streamlining the sample generation process. This paper discusses the convergence properties of FM for large sample size under the p-Wasserstein distance, a measure of distributional discrepancy. We establish that FM can achieve an almost minimax optimal convergence rate for 1 ≤ p ≤ 2, presenting the first theoretical evidence that FM can reach convergence rates comparable to those of diffusion models. Our analysis extends existing frameworks by examining a broader class of mean and variance functions for the vector fields and identifies specific conditions necessary to attain almost optimal rates.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 28 of its 35 references that are in the map. First posted on arXiv 31 May 2024.

Builds on

28 of 35 references.

Closest ancestors · 10

Same direction, newest first

  1. 2024Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation30 Apr
  2. 2024Convergence Analysis of Flow Matching in Latent Space with Transformers3 Apr
  3. 2024Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation29 Feb
  4. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generation3 Oct
  5. 2023Equivariant flow matching26 Jun
  6. 2023Functional Flow Matching26 May
  7. 2023Improving and generalizing flow-based generative models with minibatch optimal transport1 Feb
  8. 2022Flow Matching for Generative Modeling6 Oct
  9. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep
  10. 2022Equivariant Diffusion for Molecule Generation in 3D31 Mar

Other references · 18

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  2. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesiscited by 4,094
  3. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  4. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  5. 2023Voicebox: Text-Guided Multilingual Universal Speech Generation at Scalecited by 373
  6. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  7. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 215
  8. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  9. 20233D Equivariant Diffusion for Target-Aware Molecule Generation and Affinity Predictioncited by 176
  10. 2018Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionalitycited by 160
  11. 2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 159
  12. 2023Stochastic interpolants with data-dependent couplingscited by 94
6 more
  1. 2023Error Bounds for Flow Matching Methodscited by 88
  2. 2019Deep ReLU network approximation of functions on a manifoldcited by 73
  3. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptionscited by 45
  4. 2023Motion Flow Matching for Human Motion Synthesis and Editingcited by 36
  5. 2019On the minimax optimality and superiority of deep neural network learning over sparse parameter spacescited by 24
  6. 2023Multimarginal generative modeling with stochastic interpolantscited by 17

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 13; 2026: 12.

27 papers.

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  1. 2025High-Order Matching for One-Step Shortcut Diffusion Modelscited by 15
  2. 2025A Theoretical Analysis of Discrete Flow Matching Generative Modelscited by 13
  3. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flowscited by 9
  4. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 9
  5. 2026Flow Matching is Adaptive to Manifold Structurescited by 8
  6. 2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 5
  7. 2025On Flow Matching KL Divergencecited by 5
  8. 2024Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergencescited by 3
  9. 2025Iterative Flow Matching: Path Correction and Gradual Refinement for Enhanced Generative Modelingcited by 3
  10. 2026RMFlow: Refined Mean Flow by a Noise-Injection Step for Multimodal Generationcited by 3
  11. 2024Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transportcited by 2
  12. 2025Theoretical Foundation of Flow-Based Time Series Generation: Provable Approximation, Generalization, and Efficiencycited by 2
15 more
  1. 2025On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reductioncited by 2
  2. 2025Approximation Capabilities of Feedforward Neural Networks with GELU Activationscited by 2
  3. 2026Query Lower Bounds for Diffusion Samplingcited by 2
  4. 2025From Conditional to Unconditional Independence: Testing Conditional Independence via Transport Mapscited by 1
  5. 2025Graph Flow Matching: Enhancing Image Generation with Neighbor-Aware Flow Fieldscited by 1
  6. 2026First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systemscited by 1
  7. 2026Discretization and Statistical Consistency of Functional Flow Matchingcited by 1
  8. 2025Risk-Entropic Flow Matchingcited by 0
  9. 2026Posterior Augmented Flow Matchingcited by 0
  10. 2026Support-Conditioned Flow Matching Is Kernel Smoothingcited by 0
  11. 2026A Theory on Flow Matching with Neural Networkscited by 0
  12. 2026Logit-Coordinate Generative Models for Mixed Continuous-Categorical Tabular Datacited by 0
  13. 2026Population-Level Generative Modeling for Ranking Datacited by 0
  14. 2026Learning a Flow to Self-Supervised Representationscited by 0
  15. 2026Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Supportcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions
  2. 2024Theoretical guarantees in KL for Diffusion Flow Matching
  3. 2026Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees
  4. 2025On The Hidden Biases of Flow Matching Samplers
  5. 2024Local Flow Matching Generative Models
  6. 2025Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching
  7. 2026Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities
  8. 2025Demystifying Transition Matching: When and Why It Can Beat Flow Matching