Flow matching achieves minimax optimal convergence
<|reference_start|>Flow matching achieves minimax optimal convergence: 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 the minmax 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.<|reference_end|>
@article{fukumizu2024flow,
    title={Flow matching achieves almost minimax optimal convergence},
    author={Kenji Fukumizu, Taiji Suzuki, Noboru Isobe, Kazusato Oko, Masanori
  Koyama},
    journal={arXiv preprint arXiv:2405.20879},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.20879},
    primaryClass={cs.LG}
}