A genericity property of Fr\'{e}chet sample means on Riemannian manifolds

TL;DR

This paper proves that for data sampled from a continuous distribution on a Riemannian manifold, the Fréchet mean almost surely avoids any measure-zero subset, establishing a strong genericity property for these means.

Contribution

The paper introduces a general geometric theorem showing Fréchet means almost surely do not lie in measure-zero subsets, extending to equivariant means on quotient manifolds.

Findings

Fréchet means almost surely avoid measure-zero sets in the manifold.

The results apply to equivariant Fréchet means on quotient manifolds.

Application demonstrated for partial scaling-rotation means of matrices.

Abstract

Let $(M,g)$ be a Riemannian manifold. If $\mu$ is a probability measure on $M$ given by a continuous density function, one would expect the Fr\'{e}chet means of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $\mu$, to behave ``generically''; e.g. the probability that the Fr\'{e}chet mean set $\mbox{FM}(Q)$ has any elements that lie in a given, positive-codimension submanifold, should be zero for any $N\geq 1$. Even this simplest instance of genericity does not seem to have been proven in the literature, except in special cases. The main result of this paper is a general, and stronger, genericity property: given i.i.d. absolutely continuous $M$-valued random variables $X_1,\dots, X_N$, and a subset $A\subset M$ of volume-measure zero, $\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$ We also establish a companion theorem for equivariant Fr\'{e}chet means, defined when $(M,g)$ arises as the quotient of a Riemannian manifold $(\widetilde{M},\tilde{g})$ by a free, isometric action of a finite group. The equivariant Fr\'{e}chet means lie in $\widetilde{M}$, but, as we show, project down to the ordinary Fr\'{e}chet sample means, and enjoy a similar genericity property. Both these theorems are proven as consequences of a purely geometric (and quite general) result that constitutes the core mathematics in this paper: If $A\subset M$ has volume zero in $M$ , then the set $\{Q\in M^N : \mbox{FM}(Q) \cap A\neq\emptyset\}$ has volume zero in $M^N$. We conclude the paper with an application to partial scaling-rotation means, a type of mean for symmetric positive-definite matrices.