Generalized Wasserstein Barycenters

TL;DR

This paper investigates the properties of Wasserstein barycenters for signed probability measures in Hilbert spaces, including existence, uniqueness, stability, and convergence, with specific focus on one-dimensional cases and counterexamples in higher dimensions.

Contribution

It extends the theory of Wasserstein barycenters to signed measures, providing new results on existence, uniqueness, stability, and convergence, especially in one-dimensional settings.

Findings

Existence of barycenters for signed measures in Hilbert spaces.

Uniqueness in the case where the positive part is atomic.

Counterexample to uniqueness in two-dimensional space.

Abstract

We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is atomic, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the $L^2$-barycenter of the quantiles on the cone of nonincreasing functions in $L^2(0,1)$. Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in $\mathbb{R}^2$. Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure.