Kantorovich-Rubinstein distance and barycenter for finitely supported measures: Foundations and Algorithms

TL;DR

This paper introduces a systematic study of Kantorovich-Rubinstein barycenters and distances for finitely supported measures using unbalanced optimal transport, detailing their properties, explicit solutions, and computational approaches.

Contribution

It provides a detailed analysis of KR barycenters, including structural properties, explicit solutions on ultra-metric trees, and existence of sparse barycenters, expanding the understanding of unbalanced optimal transport.

Findings

KR barycenters have finite support determined by input measures.

Closed-form solutions for KR distance on ultra-metric trees.

KR barycenters outperform OT barycenters on synthetic datasets.

Abstract

The purpose of this paper is to provide a systematic discussion of a generalized barycenter based on a variant of unbalanced optimal transport (UOT) that defines a distance between general non-negative, finitely supported measures by allowing for mass creation and destruction modeled by some cost parameter. They are denoted as Kantorovich-Rubinstein (KR) barycenter and distance. In particular, we detail the influence of the cost parameter to structural properties of the KR barycenter and the KR distance. For the latter we highlight a closed form solution on ultra-metric trees. The support of such KR barycenters of finitely supported measures turns out to be finite in general and its structure to be explicitly specified by the support of the input measures. Additionally, we prove the existence of sparse KR barycenters and discuss potential computational approaches. The performance of the KR barycenter is compared to the OT barycenter on a multitude of synthetic datasets. We also consider barycenters based on the recently introduced Gaussian Hellinger-Kantorovich and Wasserstein-Fisher-Rao distances.