2023/09/25 by David Groisser, Sungkyu Jung, Groisser, David +3 · 1 citation
Computer Science · #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2309.13823
Let (M,g) be a Riemannian manifold. If μ is a probability measure on M given by a continuous density function, one would expect the Fréchet means of data-samples Q=(q1,q2,…, qN)∈ MN, with respect to μ, to behave ``generically''; e.g. the probability that the Fréchet mean set FM(Q) has any elements that lie in a given, positive-codimension submanifold, should be zero for any N≥ 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 X1,…, XN, and a subset A⊂ M of volume-measure zero, Pr\FM(\X1,…,XN\)⊂ M\backslash A\=1. We also establish a companion theorem for equivariant Fréchet means, defined when (M,g) arises as the quotient of a Riemannian manifold (\widetildeM,g) by a free, isometric action of a finite group. The equivariant Fréchet means lie in \widetildeM, but, as we show, project down to the ordinary Fréchet 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⊂ M has volume zero in M , then the set \Q∈ MN : FM(Q) ∩ A≠∅\ has volume zero in MN. We conclude the paper with an application to partial scaling-rotation means, a type of mean for symmetric positive-definite matrices.