Flow Matching has recently gained attention in generative modeling as a simple and flexible alternative to diffusion models. 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 achieves the optimal rate up to logarithmic factors. If the target distribution lies on a lower-dimensional manifold, we show that the kernel density estimator profits from the smaller intrinsic dimension on a small tube around the manifold. The faster rate also applies to Flow Matching, providing a theoretical foundation for its empirical success in high-dimensional settings.

翻译：流匹配最近在生成建模领域受到关注，作为一种简单而灵活的扩散模型替代方法。虽然现有的统计保证采用了扩散模型分析中的工具，但我们通过将流匹配与核密度估计联系起来，采取了不同的视角。我们首先验证了核密度估计器在Wasserstein距离下达到最优收敛速率（至对数因子），改进了高斯核的现有界限。基于这一结果，我们证明对于足够大的网络，流匹配能够达到至对数因子的最优速率。如果目标分布位于低维流形上，我们表明核密度估计器在流形周围的小管状区域内受益于较小的本征维度。这一更快的速率同样适用于流匹配，为其在高维设置中的实证成功提供了理论基础。