On a linearization of quadratic Wasserstein distance

Abstract

This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W 2 . In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = −∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1] 2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented.