Barycenters in the Hellinger-Kantorovich space

TL;DR

This paper proves the existence and uniqueness of Hellinger-Kantorovich barycenters in certain metric spaces, and explores their relation to multimarginal problems, extending concepts similar to Wasserstein barycenters.

Contribution

It establishes existence and uniqueness results for Hellinger-Kantorovich barycenters in compact and Polish $CAT(1)$ spaces, and introduces multimarginal problems related to these barycenters.

Findings

Existence of barycenters in compact and Polish $CAT(1)$ spaces.

Uniqueness of barycenters under additional conditions.

Relations between multimarginal solutions and barycenters.

Abstract

Recently, Liero, Mielke and Savar\'{e} introduced Hellinger-Kantorovich distance on the space of nonnegative Radon measures of a metric space $X$ [19,20]. We prove that Hellinger-Kantorovich barycenters always exist for a class of metric spaces containing of compact spaces, and Polish $CAT(1)$ spaces; and if we assume further some conditions on starting measures, such barycenters are unique. We also introduce homogeneous multimarginal problems and illustrate some relations between their solutions with Hellinger-Kantorovich barycenters. Our results are analogous to the work of Agueh and Carlier [1] for Wassertein barycenters.