Flow matching achieves almost minimax optimal convergence
Kenji Fukumizu Taiji Suzuki Noboru Isobe Kazusato Oko Masanori Koyama
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.
flow matching, generative model, convergence rate, optimality
We establish that FM can achieve an almost minmax optimal convergence rate in terms of 2-Wasserstein distance.