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arXiv · 2201.13386

On a linearization of quadratic Wasserstein distance

Philip Greengard·Jeremy G. Hoskins·Nicholas F. Marshall·Amit Singer
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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.

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Philip Greengard, Jeremy G. Hoskins, Nicholas F. Marshall, Amit Singer. 2022-01-31. On a linearization of quadratic Wasserstein distance. https://arxiv.org/abs/2201.13386

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