Flow matching achieves almost minimax optimal convergence

Kenji Fukumizu, Taiji Suzuki, Noboru Isobe, Kazusato Oko, Masanori Koyama

Keywords: flow matching, generative model, convergence rate, optimality

TL;DR: We establish that FM can achieve an almost minmax optimal convergence rate in terms of 2-Wasserstein distance.

Abstract: 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 in terms of the $p$-Wasserstein distance, a measure of distributional discrepancy. We establish that FM can achieve an almost minimax optimal convergence rate for $1 \leq p \leq 2$, 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 these optimal rates.

Primary Area: generative models

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Submission Number: 9156