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 an elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form $H=-\hat{I}”+V$, and describe an associated embedding. Third, for the case of probability distributions on the unit square $[0,1{]}^2$ represented by $n\times n$ arrays we present a fast code demonstrating our approach. Several numerical examples are presented.

24 pages, 6 figures