Necessary and sufficient condition for the existence of a Fr\'echet mean on the circle

TL;DR

This paper establishes a precise condition for the existence of a Fréchet mean on the circle, extending previous results and providing an algorithm for computation and convergence analysis.

Contribution

It provides a necessary and sufficient condition for the existence of the Fréchet mean on the circle, including a new support-free criterion and convergence results.

Findings

Characterization of when a Fréchet mean exists on the circle

Introduction of a new support-free existence condition

Algorithm for computing the Fréchet mean and analyzing convergence

Abstract

Let $(\S^1,d_{\S^1})$ be the unit circle in $\R^2$ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure $\mu$ to admit a well defined Fr\'echet mean on $(\S^1,d_{\S^1})$. %This criterion allows to recover already known sufficient conditions of existence. We derive a new sufficient condition of existence $P(\alpha,\varphi)$ with no restriction on the support of the measure. Then, we study the convergence of the empirical Fr\'echet mean to the Fr\'echet mean and we give an algorithm to compute it.