Riemannian barycentres of Gibbs distributions: new results on concentration and convexity in compact symmetric spaces

Abstract

The Riemannian barycentre (or Fréchet mean) is the workhorse of data analysis for data taking values in Riemannian manifolds. The Riemannian barycentre of a probability distribution P on a Riemannian manifold M is a possible generalisation of the concept of expected value, at least when the barycentre is unique. Knowing when the barycentre of P is unique is of fundamental importance for its interpretation and computation. Existing results can only guarantee this uniqueness by assuming P is supported inside a convex geodesic ball B(x∗, δ) ⊂ M. This assumption is overly restrictive since many distributions have support equal to M yet are sufficiently concentrated within a convex geodesic ball ..