On a linearization of quadratic Wasserstein distance

TL;DR

This paper introduces a linear approximation method for quadratic Wasserstein distance using Sobolev norms, involving elliptic PDEs, with practical implementation and numerical validation.

Contribution

It provides a new linearization approach for Wasserstein distance, connecting it to Sobolev norms, and offers computational techniques and numerical demonstrations.

Findings

Quantitative error estimates for the linearization.

Reduction to solving elliptic boundary value problems.

Fast implementation for 2D probability distributions.

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-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 $H = -\Delta + V$, 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.