On a linearization of quadratic Wasserstein distance

01/31/2022

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by   Philip Greengard, et al.

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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. econd, 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 = -Δ + 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.