Flow matching achieves almost minimax optimal convergence

TL;DR

This paper provides the first theoretical analysis showing that flow matching, a simpler alternative to diffusion models, can achieve near-minimax optimal convergence rates for large sample sizes under the $p$-Wasserstein distance.

Contribution

It establishes the convergence properties of flow matching, demonstrating its potential to match the performance of diffusion models in terms of convergence rates.

Findings

Flow matching achieves almost minimax optimal convergence for $1 \,\leq p \leq 2$.

Theoretical evidence that flow matching can reach convergence rates comparable to diffusion models.

Broader class of mean and variance functions analyzed for convergence conditions.

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 for large sample size under 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 almost optimal rates.