In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability $L^\infty$ and approximate $\mathcal{C}^{0,1}$ convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the $L^2$-convergence rates established by two of the authors in previous work.

翻译：在本文中,我们从随机数据点构建的几何图形上研究利普西茨省 PDE 的常规性。 数据点是从光滑的方块支持的分布中抽样的。 我们所研究的方程式的组合在基于图形的学习数据分析中产生, 作为重要的例子, 包含由Laplacian egenvestors图所满足的方程式。 特别是, 我们证明, 用于解决Poisson 方程式的图解, 内部和全球利普西茨估计值的概率很高。 我们的结果可以用来显示, Laplacian egenvectors, 极有可能是Lipschitz 的常规, 与恒定的常数一致, 明显取决于相应的电子值。 我们的分析依赖于一个在连续水平上合适的随机点云函数的概率组合, 以及一个将随机点云的功能延伸至连续方块数的中间法方法。 作为我们一般定期性结果的一个副产品, 我们得到了高的概率 $\fncal, 和近 $\cal=cal0.1} 美元, 。