The Signed Wasserstein Barycenter Problem

TL;DR

This paper investigates the properties of the signed Wasserstein barycenter problem, focusing on solution uniqueness and duality, especially when negative weights are involved, which complicates the convexity of the problem.

Contribution

It extends the uniqueness results for signed barycenters to general costs with one positive weight and introduces a dual formulation with conditions for optimality in the general case.

Findings

Established uniqueness of solutions with a single positive weight.

Introduced a dual problem framework using Kantorovich potentials.

Provided sufficient conditions for stationary solutions to be optimal.

Abstract

Barycenter problems encode important geometric information about a metric space. While these problems are typically studied with positive weight coefficients associated to each distance term, more general signed Wasserstein barycenter problems have recently drawn a great deal of interest. These mixed sign problems have appeared in statistical inference setting as a way to generalize least squares regression to measure valued outputs and have appeared in numerical methods to improve the accuracy of Wasserstein gradient flow solvers. Unfortunately, the presence of negatively weighted distance terms destroys the Euclidean convexity of the unsigned problem, resulting in a much more challenging optimization task. The main focus of this work is to study properties of the signed barycenter problem for a general transport cost with a focus on establishing uniqueness of solutions. In particular, when there is only one positive weight, we extend the uniqueness result of Tornabene et al. (2025) to any cost satisfying a certain convexity property. In the case of arbitrary weights, we introduce the dual problem in terms of Kantorovich potentials and provide a sufficient condition for a stationary solution of the dual problem to induce an optimal signed barycenter.