On the Uniqueness of Fr\'echet Means for Polytope Norms

TL;DR

This paper investigates the conditions under which Fréchet means are unique in polytope normed spaces, providing geometric characterizations and thresholds related to sample size, especially for $\,ell_\,infty$ and $\,ell_\,1$ norms.

Contribution

It offers a geometric analysis of Fréchet mean uniqueness in polytope normed spaces and determines the sample size threshold for guaranteed uniqueness.

Findings

Threshold sample size for uniqueness is at most one more than the space dimension.

Derived geometric conditions for the probability of Fréchet mean uniqueness.

Computed explicit thresholds for $\,ell_\,infty$ and $\,ell_\,1$ norms.

Abstract

Fr\'echet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fr\'echet means on normed spaces whose unit ball is a polytope; this setting is rarely covered by existing literature on Fr\'echet means, which focuses on smooth spaces or spaces with bounded curvature. We study the geometry of the set of Fr\'echet means over polytope normed spaces, with a focus on dimension and probabilistic conditions for uniqueness. In particular, we provide a geometric characterisation of the threshold sample size at which Fr\'echet means have a positive probability of being unique, and we prove that this threshold is at most one more than the dimension of our space. We are able to use this geometric characterisation to compute the unique Fr\'echet mean sample threshold in the case of the $\ell_\infty$ and $\ell_1$ norms.