Quadratically Regularized Optimal Transport: nearly optimal potentials and convergence of discrete Laplace operators

TL;DR

This paper investigates quadratic regularization in optimal transport to construct graphs whose discrete Laplace operators converge to the Laplace--Beltrami operator, providing theoretical insights and simulation validation.

Contribution

It derives first order optimal potentials for quadratic regularized optimal transport and proves convergence of the associated discrete Laplace operators to the continuous Laplace--Beltrami operator.

Findings

Optimal potentials resemble Barenblatt--Prattle solutions.

Discrete Laplace operators converge in $L^2$ to the continuous operator.

Simulation results support theoretical convergence.

Abstract

We consider the conjecture proposed in Matsumoto, Zhang and Schiebinger (2022) suggesting that optimal transport with quadratic regularisation can be used to construct a graph whose discrete Laplace operator converges to the Laplace--Beltrami operator. We derive first order optimal potentials for the problem under consideration and find that the resulting solutions exhibit a surprising resemblance to the well-known Barenblatt--Prattle solution of the porous medium equation. Then, relying on these first order optimal potentials, we derive the pointwise $L^2$-limit of such discrete operators built from an i.i.d. random sample on a smooth compact manifold. Simulation results complementing the limiting distribution results are also presented.