Barycenters and a law of large numbers in Gromov hyperbolic spaces

TL;DR

This paper explores barycenters of probability measures in Gromov hyperbolic spaces, establishing key properties like contraction, approximation, and a law of large numbers, extending known results from CAT(0)-spaces.

Contribution

It introduces new contraction and approximation results for barycenters in Gromov hyperbolic spaces, generalizing prior work on CAT(0)-spaces.

Findings

Wasserstein distance bounds barycenter distances

Deterministic approximation of barycenters for finite point distributions

A law of large numbers for barycenters in hyperbolic spaces

Abstract

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.