Flow Matching is Adaptive to Manifold Structures

• URL: http://arxiv.org/abs/2602.22486v1
• Date: Wed, 25 Feb 2026 23:52:32 GMT
• Title: Flow Matching is Adaptive to Manifold Structures
• Authors: Shivam Kumar, Yixin Wang, Lizhen Lin,
• Abstract summary: Flow matching is a simulation-based alternative to diffusion-based generative modeling.<n>We show how flow matching adapts to data geometry and circumvents the curse of dimensionality.
• Score: 32.55405572762157
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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.

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