On the computation of the infinity Wasserstein distance and the Wasserstein Projection Problem

Computing the infinity Wasserstein distance and retrieving projections of a probability measure onto a closed subset of probability measures are critical sub-problems in various applied fields. However, the practical applicability of these objects is limited by two factors: either the associated quantities are computationally prohibitive or there is a lack of available algorithms capable of calculating them. In this paper, we propose a novel class of Linear Programming problems and a routine that allows us to compute the infinity Wasserstein distance and to compute a projection of a probability measure over a generic subset of probability measures with respect to any $p$-Wasserstein distance with $p\in[1,\infty]$.