A Brief History of Fr\'echet Distances: From Curves and Probability Laws to FID

TL;DR

This paper traces the historical development of Fréchet distances from their origins in geometry and probability to modern applications like the Fréchet Inception Distance in deep learning.

Contribution

It provides a chronological overview connecting geometric and distributional facets of Fréchet distances, highlighting their evolution and modern use cases.

Findings

Links Fréchet distances in geometry and probability laws.

Explains the modern use of FID as Wasserstein-2 distance.

Includes translations of original foundational papers.

Abstract

This note provides a chronological account of Fr\'echet distances, starting with Maurice Fr\'echet's 1906 doctoral thesis on distances in abstract sets and tracing the Fr\'echet 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 Fr\'echet distances on probability laws and the geometric facet on curves. The note ends with a modern use case, the Fr\'echet 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 Fr\'echet's 1906 thesis and 1957 paper, and L\'evy's 1950 note for reader convenience.