Hellinger-Kantorovich barycenter between Dirac measures

TL;DR

This paper investigates the properties of the Hellinger-Kantorovich barycenter for Dirac measures, revealing its complex behavior and potential for multiscale representation, supported by analytical and numerical analysis.

Contribution

It provides a detailed analysis of the HK barycenter for Dirac measures, including its dependence on parameters and its discrete or continuous nature, which was less understood before.

Findings

HK barycenter exhibits complex behavior compared to Wasserstein barycenter.

The barycenter can serve as a coarse-to-fine representation of measures.

Numerical experiments confirm the multiscale representation capability.

Abstract

The Hellinger-Kantorovich (HK) distance is an unbalanced extension of the Wasserstein-2 distance. It was shown recently that the HK barycenter exhibits a much more complex behaviour than the Wasserstein barycenter. Motivated by this observation we study the HK barycenter in more detail for the case where the input measures are an uncountable collection of Dirac measures, in particular the dependency on the length scale parameter of HK, the question whether the HK barycenter is discrete or continuous and the relation between the expected and the empirical barycenter. The analytical results are complemented with numerical experiments that demonstrate that the HK barycenter can provide a coarse-to-fine representation of an input pointcloud or measure.