A kernel formula for regularized Wasserstein proximal operators

2023/01/24 by Wuchen Li, Li, Wuchen, Siting Liu +3 · 2 citations
Engineering · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2301.10301

openalex publication_date 2023/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a class of regularized proximal operators in Wasserstein-2 space. We derive their solutions by kernel integration formulas. We obtain the Wasserstein proximal operator using a pair of forward-backward partial differential equations consisting of a continuity equation and a Hamilton-Jacobi equation with a terminal time potential function and an initial time density function. We regularize the PDE pair by adding forward and backward Laplacian operators. We apply Hopf-Cole type transformations to rewrite these regularized PDE pairs into forward-backward heat equations. We then use the fundamental solution of the heat equation to represent the regularized Wasserstein proximal with kernel integral formulas. Numerical examples show the effectiveness of kernel formulas in approximating the Wasserstein proximal operator.

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