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