Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces

This paper is concerned with the problem of defining and estimating statistics for distributions on spaces such as Riemannian manifolds and more general metric spaces. The challenge comes, in part, from the fact that statistics such as means and modes may be unstable: for example, a small perturbation to a distribution can lead to a large change in Fr\'echet means on spaces as simple as a circle. We address this issue by introducing a new merge tree representation of barycenters called the barycentric merge tree (BMT), which takes the form of a measured metric graph and summarizes features of the distribution in a multiscale manner. Modes are treated as special cases of barycenters through diffusion distances. In contrast to the properties of classical means and modes, we prove that BMTs are stable -- this is quantified as a Lipschitz estimate involving optimal transport metrics. This stability allows us to derive a consistency result for approximating BMTs from empirical measures, with explicit convergence rates. We also give a provably accurate method for discretely approximating the BMT construction and use this to provide numerical examples for distributions on spheres and shape spaces.