Flow Matching理论最优性获证明，与核密度估计关联突破
On the minimax optimality of Flow Matching through the connection to kernel density estimation
- 将Flow Matching与核密度估计关联，构建新分析框架
- 证明大网络下可达到近似最优收敛率（含对数因子）
- 解释其在高维数据中表现优异的理论原因
Flow Matching近年来作为生成模型的简洁灵活替代方案受到关注。现有统计保证多借鉴扩散模型分析工具，本文另辟蹊径，将其与核密度估计相联系。首先验证核密度估计在Wasserstein距离下的收敛率可达最优（对数因子内），优于已有高斯核界。基于此，证明当网络足够大时，Flow Matching亦能达到近似最优收敛率。若目标分布位于低维流形上，核密度估计可在流形附近小管邻域内利用更低的内在维度获得更快收敛率；该加速效应同样适用于Flow Matching，为其实验成功提供了理论基础，尤其在高维设置中。
原文摘要 · Abstract (English)
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.
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