A kernel formula for regularized Wasserstein proximal operators

TL;DR

This paper introduces a kernel integral formula for regularized Wasserstein-2 proximal operators, derived via PDE transformations and heat kernel methods, enabling efficient approximation and computation.

Contribution

It presents a novel kernel formula for Wasserstein proximal operators using PDE regularization and heat kernel techniques, advancing computational methods in optimal transport.

Findings

Kernel formulas effectively approximate Wasserstein proximal operators.

Regularized PDE approach simplifies computation of Wasserstein operators.

Numerical examples demonstrate the accuracy of the kernel-based method.

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.