Title: Quadratically Regularized Optimal Transport: nearly optimal potentials and convergence of discrete Laplace operators Show Chinese title

Title: 二次正则化最优传输：几乎最优势函数和离散拉普拉斯算子的收敛性

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. Show Chinese abstract

Abstract: 我们考虑Matsumoto、Zhang和Schiebinger（2022）提出的猜想，该猜想表明可以使用带有二次正则化的最优传输来构建一个图，其离散拉普拉斯算子收敛于拉普拉斯-贝尔特拉米算子。 我们推导了所考虑问题的一阶最优势，并发现所得解与多孔介质方程的著名Barenblatt-Prattle解表现出惊人的相似性。 然后，依赖于这些一阶最优势，我们推导了从光滑紧致流形上的独立同分布随机样本构建的此类离散算子的逐点$L^2$极限。 模拟结果也展示了极限分布结果。