On the minimax optimality of Flow Matching through the connection to kernel density estimation

Abstract

Flow Matching has recently gained attention in generative modeling as a simple and flexible alternative to diffusion models, the current state of the art. While existing statistical guarantees adapt tools from the analysis of diffusion models, we take a different perspective by connecting Flow Matching to kernel density estimation. We first verify that the kernel density estimator matches the optimal rate of convergence in Wasserstein distance up to logarithmic factors, improving existing bounds for the Gaussian kernel. Based on this result, we prove that for sufficiently large networks, Flow Matching also achieves the optimal rate up to logarithmic factors, providing a theoretical foundation for the empirical success of this method. Finally, we provide a first justification of Flow Matching's effectiveness in high-dimensional settings by showing that rates improve when the target distribution lies on a lower-dimensional linear subspace.