On a linearization of quadratic Wasserstein distance
Abstract
This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W2. 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-Amp\'ere 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 Schr\"odinger 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.