Flow Matching is Adaptive to Manifold Structures

TL;DR

This paper provides a theoretical analysis of flow matching on data supported on smooth manifolds, explaining its effectiveness and convergence properties in low-dimensional structures within high-dimensional spaces.

Contribution

It establishes non-asymptotic convergence guarantees for flow matching with manifold-supported targets, highlighting adaptation to intrinsic data geometry.

Findings

Proves near minimax-optimal convergence rate depending on intrinsic dimension

Demonstrates flow matching's effectiveness in manifold-supported data settings

Provides theoretical justification for empirical success in high-dimensional tasks

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.