Linear programming for Wasserstein barycenters

This paper presents a family of generative Linear Programming models for the computation of the Wasserstein Barycenter of a large set of two-dimensional images. Wasserstein Barycenters were recently introduced to mathematically generalize the concept of averaging a set of points, to the concept of averaging a set of clouds of points, such as, for instance, two-dimensional images. In Machine Learning terms, the Wasserstein Barycenter problem is a generative constrained optimization problem, since the values of the decision variables of the optimal solution give a new image that represents the average of the input images. Our family of Linear Programming models rely on different types of Kantorovich-Wasserstein distances used to compute a barycenter, and they are efficiently solved with a modern commercial Linear Programming solver. We numerically show the strength of the proposed models by computing and plotting the barycenters of all digits included in the classical MNIST dataset.