Zero Probability of the Cut Locus of a Fr\'echet Mean on a Riemannian Manifold

TL;DR

This paper proves that the probability of the cut locus of a Fréchet mean on a Riemannian manifold is zero, extending previous results and addressing implications for statistical analysis on manifolds.

Contribution

It establishes the zero probability of the cut locus for Fréchet means in general Riemannian manifolds, generalizing earlier special case results.

Findings

Cut locus of a Fréchet mean has zero probability

Ruling out stickiness in statistical analysis

Examples of nowhere smooth Fréchet functions

Abstract

We show that the cut locus of a Fr\'echet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fr\'echet functions and we discuss extensions of the statement to Fr\'echet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general metric spaces.