Sample-Based Optimal Transport and Barycenter Problems

A methodology is developed for the numerical solution to the sample-based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets.