2022/01/31 by Philip Greengard, Greengard, Philip, Jeremy Hoskins +5
Mathematics · Medicine · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hidradenitis Suppurativa and Treatments #Numerical Analysis (math.NA) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2201.13386
openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-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 = -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.