The Signed Wasserstein Barycenter Problem
Matt Jacobs, Bohan Zhou
TL;DR
This work develops the theory of the signed Wasserstein barycenter for general cost functions with mixed positive and negative weights. It proves existence under broad cost-growth assumptions and develops a dual framework that yields stationary, saddle-point, and optimality conditions, connecting dual Potentials to primal barycenters via generalized transport maps. A key contribution is a convexity result for the single-positive-weight case, which extends $\,\lambda$-convexity along generalized geodesics beyond the quadratic cost. The proposed duality-based approach also provides a sufficient condition to upgrade stationary points to saddle points, enabling practical algorithms for computing signed barycenters and paving the way for distribution-to-distribution regression and high-order Wasserstein gradient flows. Overall, the paper broadens the scope of barycenter analysis to general costs, elucidates dual structures, and establishes foundations for robust numerical methods in nonconvex OT settings.
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.