Kantorovich–Rubinstein Distance and Barycenter for Finitely Supported Measures: Foundations and Algorithms

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.

Unbalanced Kantorovich-Rubinstein distance and barycenter for finitely supported measures: A statistical perspective

“…Following Heinemann et al [2022a], for p ≥ 1 and a parameter C > 0, the (p, C)−Kantorovich-Rubinstein distance (KRD) between two measures µ, ν ∈ M + (X ) is defined as (2)…”

“…(b) The UOT plan for the (2, 2)-KRD between the two unnormalised measures (all points have mass 1). 2018b, Balaji et al, 2020, Mukherjee et al, 2021, Heinemann et al, 2022a. These formulations extend optimal transport concepts to general positive measures by either fixing the total amount of mass to be transported in advance or by penalising the hard marginal constraints inherent in OT.…”

“…These formulations extend optimal transport concepts to general positive measures by either fixing the total amount of mass to be transported in advance or by penalising the hard marginal constraints inherent in OT. All these approaches also give rise to associated barycenters, generalising the popular notion of OT barycenters [Agueh and Carlier, 2011] to measures of unequal total intensity [Chizat et al, 2018a, Friesecke et al, 2021, Heinemann et al, 2022a. The UOT formulation considered here is the (p, C)-Kantorovich-Rubinstein distance (KRD) (see Section 1.1) whose structural properties and related (p, C)-barycenter have recently been studied in detail [Heinemann et al, 2022a].…”

“…All these approaches also give rise to associated barycenters, generalising the popular notion of OT barycenters [Agueh and Carlier, 2011] to measures of unequal total intensity [Chizat et al, 2018a, Friesecke et al, 2021, Heinemann et al, 2022a. The UOT formulation considered here is the (p, C)-Kantorovich-Rubinstein distance (KRD) (see Section 1.1) whose structural properties and related (p, C)-barycenter have recently been studied in detail [Heinemann et al, 2022a]. The (1, 1)-KRD essentially corresponds to the notion of extended Kantorovich norms [Kantorovich andRubinstein, 1958, Hanin, 1992] considered in the context of Lipschitz spaces and signed measures.…”

“…From a data analysis point of view we find it particularly appealing that for the (p, C)-KRD there is a clear geometrical connection between its penalty C and the structural properties of the corresponding UOT plans and (p, C)-barycenter. More precisely, it is shown in Heinemann et al [2022a] that C controls the largest scale at which mass transport is possible in an optimal plan. This interpretation of the (p, C)-KRD allows to easily design it to respect different structural properties of the data and thus makes it a prime candidate for statistical tasks in data analysis.…”