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

Overview & Synopsis

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.