Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers

TL;DR

This paper provides a rigorous theoretical analysis of flow matching ODE models, revealing how denoisers influence dynamics in relation to data geometry, and establishing convergence and stability properties.

Contribution

It introduces a comprehensive theory of FM ODEs, analyzing sample trajectories and their interaction with data geometry, including convergence on low-dimensional manifolds.

Findings

Denoisers guide ODE dynamics through attraction and absorption behaviors.

Proves convergence of FM ODEs under weak assumptions, even on low-dimensional manifolds.

Provides insights into memorization phenomena and equivariance properties of FM ODEs.

Abstract

Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold-cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.