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 ∈ [ 1 , ∞ ] . • We propose a novel algorithm capable of computing the infinity Wasserstein distance. • We use the algorithm to retrieve a projection of a measure on suitable subsets of Wasserstein spaces. • We build a novel bound on the gap between the Wasserstein distance and its truncated counterpart.