论文标题
二次正规化最佳运输：离散拉普拉斯操作员的几乎最佳电位和收敛性
Quadratically Regularized Optimal Transport: nearly optimal potentials and convergence of discrete Laplace operators
论文作者
论文摘要
我们考虑了Matsumoto，Zhang和Schiebinger（2022）中提出的猜想，这表明可以使用二次正则化的最佳传输来构建一个图形，其离散的Laplace操作员会收敛到Laplace-Beltrami操作员。我们为所考虑的问题提供了一阶最佳潜力，并发现所得的解决方案与众所周知的Barenblatt（多孔介质方程）的溶液表现出令人惊讶的相似之处。然后，依靠这些一阶最佳电位，我们得出了从I.I.D.构建的此类离散操作员的端点$ l^2 $的限制。在平滑的紧凑型歧管上随机样品。还提出了补充限制分布结果的仿真结果。
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.
