Barycenters for the Hellinger--Kantorovich distance over $\mathbb{R}^d$

TL;DR

This paper investigates the properties of barycenters under the Hellinger--Kantorovich metric for non-negative measures, revealing unique behaviors and applications in analyzing Gaussian mixtures.

Contribution

It establishes existence, uniqueness, and equivalence of formulations for HK barycenters, and explores their behavior between Dirac measures with practical applications.

Findings

HK barycenter exhibits local clustering behavior

A 1-parameter family of barycenters captures scale variations

Application to Gaussian mixture analysis infers underlying components

Abstract

We study the barycenter of the Hellinger--Kantorovich metric over non-negative measures on compact, convex subsets of $\mathbb{R}^d$. The article establishes existence, uniqueness (under suitable assumptions) and equivalence between a coupled-two-marginal and a multi-marginal formulation. We analyze the HK barycenter between Dirac measures in detail, and find that it differs substantially from the Wasserstein barycenter by exhibiting a local `clustering' behaviour, depending on the length scale of the input measures. In applications it makes sense to simultaneously consider all choices of this scale, leading to a 1-parameter family of barycenters. We demonstrate the usefulness of this family by analyzing point clouds sampled from a mixture of Gaussians and inferring the number and location of the underlying Gaussians.