Flow matching achieves almost minimax optimal convergence
Kenji Fukumizu, Taiji Suzuki, Noboru Isobe, Kazusato Oko, Masanori Koyama
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.
BibTeX
@inproceedings{
fukumizu2025flow,
title={Flow matching achieves almost minimax optimal convergence},
author={Kenji Fukumizu and Taiji Suzuki and Noboru Isobe and Kazusato Oko and Masanori Koyama},
booktitle={The Thirteenth International Conference on Learning Representations},
year={2025},
url={https://openreview.net/forum?id=2OMyAFjiJJ}
}