This note provides a chronological account of Fréchet distances, starting with Maurice Fréchet's 1906 doctoral thesis on distances in abstract sets and tracing the Fréchet distance between polygonal curves and its algorithmic computation in the 1990s. It then continues with his 1957 paper on a coupling-based distance between probability laws with a brief glimpse of Wasserstein distance and optimal transport. We further attempt to draw connections between the distributional, coupling-based facet of Fréchet distances on probability laws and the geometric facet on curves. The note ends with a modern use case, the Fréchet Inception Distance (FID) in the era of deep generative model evaluation, interpretable as the Wasserstein-2 distance between multivariate Gaussians in a learned feature space. An appendix includes \TeX{}ified faithful English translations of Fréchet's 1906 thesis and 1957 paper, and Lévy's 1950 note for reader convenience.

翻译：本文按时间顺序梳理了Fréchet距离的发展历程，始于莫里斯·弗雷歇（Maurice Fréchet）1906年关于抽象集合中距离的博士论文，继而追踪了折线间的Fréchet距离及其在1990年代的算法计算。接着，文章讨论了他1957年基于耦合的概率律距离论文，并简要介绍了Wasserstein距离与最优传输理论。我们进一步尝试建立概率律上基于耦合分布的Fréchet距离与曲线几何方面之间的关联。本文以现代应用案例作为结尾：深度生成模型评估时代的Fréchet初始距离（FID），该距离可解释为学习特征空间中多元高斯分布之间的Wasserstein-2距离。附录附有Fréchet 1906年论文、1957年论文以及Lévy 1950年笔记的\TeX{}\ 化忠实英文译本，以方便读者查阅。