# Flow Matching is Adaptive to Manifold Structures arXiv: https://arxiv.org/abs/2602.22486 · HTML: https://mlsanity.com/paper/2602.22486/ ## Key facts - Cited by 10 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 26 of its 37 references that are in the map. - First posted on arXiv 25 Feb 2026. ## 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, 26 of 26) - [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) - [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017) - [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) - [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024) - [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024) - [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023) - [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019) - [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023) - [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023) - [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024) - [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024) - [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024) - [Latent Space Editing in Transformer-Based Flow Matching](https://mlsanity.com/paper/2312.10825/) (2023) - [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024) - [Convergence of Continuous Normalizing Flows for Learning Probability Distributions](https://mlsanity.com/paper/2404.00551/) (2024) - [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023) - [α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models](https://mlsanity.com/paper/2504.10283/) (2025) - [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023) - [A Theoretical Analysis of Discrete Flow Matching Generative Models](https://mlsanity.com/paper/2509.22623/) (2025) - [On the Convergence and Straightness of Rectified Flow](https://mlsanity.com/paper/2410.14949/) (2024) - [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025) - [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025) ## Built on by (most cited first, 10 of 10) - [Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2610.02663/) (2026) - [GeoFunFlow-3D: A Physics-Guided Generative Flow Matching Framework for High-Fidelity 3D Aerodynamic Inference over Complex Geometries](https://mlsanity.com/paper/2604.23350/) (2026) - [Flow Matching with Arbitrary Auxiliary Paths](https://mlsanity.com/paper/2605.06364/) (2026) - [Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine](https://mlsanity.com/paper/2605.20235/) (2026) - [Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers](https://mlsanity.com/paper/2607.04113/) (2026) - [Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective](https://mlsanity.com/paper/2609.06947/) (2026) - [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026) - [Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support](https://mlsanity.com/paper/2609.38918/) (2026) - [Sample complexity bounds for categorical Markov random fields via Discrete Diffusions](https://mlsanity.com/paper/2610.02128/) (2026) - [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026) ## Abstract (arXiv) Flow matching has emerged as a simulation-free alternative to diffusion-based generative modeling, producing samples by solving an ODE whose time-dependent velocity field is learned along an interpolation between a simple source distribution (e.g., a standard normal) and a target data distribution. Flow-based methods often exhibit greater training stability and have achieved strong empirical performance in high-dimensional settings where data concentrate near a low-dimensional manifold, such as text-to-image synthesis, video generation, and molecular structure generation. Despite this success, existing theoretical analyses of flow matching assume target distributions with smooth, full-dimensional densities, leaving its effectiveness in manifold-supported settings largely unexplained. To this end, we theoretically analyze flow matching with linear interpolation when the target distribution is supported on a smooth manifold. We establish a non-asymptotic convergence guarantee for the learned velocity field, and then propagate this estimation error through the ODE to obtain statistical consistency of the implicit density estimator induced by the flow-matching objective. The resulting convergence rate is near minimax-optimal, depends only on the intrinsic dimension, and reflects the smoothness of both the manifold and the target distribution. Together, these results provide a principled explanation for how flow matching adapts to intrinsic data geometry and circumvents the curse of dimensionality.