On the Computation of Wasserstein Barycenters

Abstract

The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. We state a general version of the equivalence of the Wasserstein barycenter problem to the n-coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n-coupling problem), also provides a new criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution a simple to implement algorithm for computing barycenters. We find in several examples that our approach has a similar time complexity with respect to well established algorithms and is able to provide an accurate and fast visualization of barycenters of a large number of measures. The proposed algorithm can also be applied to morecomplex optimization problems like the k-barycenter problem.

Keywords: Wasserstein barycenter, swapping algorithm, optimal transportations, k-means clustering, image processing