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.

翻译：我们认为Matsumoto、Zhang和Schiebinger(2022年)提出的推测表明,使用二次规范化的最佳运输方法可以用来绘制一个图,其离散拉普尔操作员与Laplace-Beltrami操作员汇合,我们为审议中的问题获得一线最佳潜力,发现由此产生的解决办法与众所周知的多孔中程方程的Barenblatt-Prattle解决方案有惊人的相似之处。然后,依靠这些第一线最佳潜力,我们从一个光滑的压合体上随机抽样中得出这种离散操作员的点值为$2美元。还介绍了补充有限分布结果的模拟结果。