# Flow matching achieves almost minimax optimal convergence

arXiv: https://arxiv.org/abs/2405.20879 · HTML: https://mlsanity.com/paper/2405.20879/

## Key facts

- Cited by 29 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 28 of its 35 references that are in the map.
- First posted on arXiv 31 May 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 28 of 28)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Voicebox: Text-Guided Multilingual Universal Speech Generation at Scale](https://mlsanity.com/paper/2306.15687/) (2023)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [3D Equivariant Diffusion for Target-Aware Molecule Generation and Affinity Prediction](https://mlsanity.com/paper/2303.03543/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation](https://mlsanity.com/paper/2404.19739/) (2024)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Motion Flow Matching for Human Motion Synthesis and Editing](https://mlsanity.com/paper/2312.08895/) (2023)
- [On the minimax optimality and superiority of deep neural network learning over sparse parameter spaces](https://mlsanity.com/paper/1905.09195/) (2019)
- [Multimarginal generative modeling with stochastic interpolants](https://mlsanity.com/paper/2310.03695/) (2023)
- [Convergence Analysis of Flow Matching in Latent Space with Transformers](https://mlsanity.com/paper/2404.02538/) (2024)
- [Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation](https://mlsanity.com/paper/2402.18839/) (2024)

## Built on by (most cited first, 29 of 29)

- [High-Order Matching for One-Step Shortcut Diffusion Models](https://mlsanity.com/paper/2502.00688/) (2025)
- [A Theoretical Analysis of Discrete Flow Matching Generative Models](https://mlsanity.com/paper/2509.22623/) (2025)
- [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025)
- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [On Flow Matching KL Divergence](https://mlsanity.com/paper/2511.05480/) (2025)
- [Iterative Flow Matching: Path Correction and Gradual Refinement for Enhanced Generative Modeling](https://mlsanity.com/paper/2502.16445/) (2025)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)
- [On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2512.18971/) (2025)
- [RMFlow: Refined Mean Flow by a Noise-Injection Step for Multimodal Generation](https://mlsanity.com/paper/2602.00849/) (2026)
- [Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport](https://mlsanity.com/paper/2407.04495/) (2024)
- [Theoretical Foundation of Flow-Based Time Series Generation: Provable Approximation, Generalization, and Efficiency](https://mlsanity.com/paper/2503.14076/) (2025)
- [From Conditional to Unconditional Independence: Testing Conditional Independence via Transport Maps](https://mlsanity.com/paper/2504.09567/) (2025)
- [Approximation Capabilities of Feedforward Neural Networks with GELU Activations](https://mlsanity.com/paper/2512.21749/) (2025)
- [Query Lower Bounds for Diffusion Sampling](https://mlsanity.com/paper/2604.10857/) (2026)
- [Graph Flow Matching: Enhancing Image Generation with Neighbor-Aware Flow Fields](https://mlsanity.com/paper/2505.24434/) (2025)
- [Posterior Augmented Flow Matching](https://mlsanity.com/paper/2605.00825/) (2026)
- [A Theory on Flow Matching with Neural Networks](https://mlsanity.com/paper/2606.10089/) (2026)
- [First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems](https://mlsanity.com/paper/2606.11138/) (2026)
- [Discretization and Statistical Consistency of Functional Flow Matching](https://mlsanity.com/paper/2608.04531/) (2026)
- [Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2610.02663/) (2026)
- [Risk-Entropic Flow Matching](https://mlsanity.com/paper/2512.03078/) (2025)
- [Support-Conditioned Flow Matching Is Kernel Smoothing](https://mlsanity.com/paper/2605.13386/) (2026)
- [Logit-Coordinate Generative Models for Mixed Continuous-Categorical Tabular Data](https://mlsanity.com/paper/2607.23348/) (2026)
- [Population-Level Generative Modeling for Ranking Data](https://mlsanity.com/paper/2608.08422/) (2026)
- [Learning a Flow to Self-Supervised Representations](https://mlsanity.com/paper/2609.29350/) (2026)
- [Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support](https://mlsanity.com/paper/2609.38918/) (2026)
- [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026)

## 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.