The Wasserstein barycenter problem with signed weights

Data Science Seminar

Matt Jacobs
UC Santa Barbara

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 L^2 convexity of the unsigned problem, resulting in a much more challenging optimization task.  In this talk, I will discuss some theoretical properties of these mixed sign barycenter problems, focusing on sufficient conditions to guarantee the global optimality and uniqueness of a critical point.