Lipschitz Regularity of Graph Laplacians on Random Data Clouds

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The points are sampled a distribution supported smooth manifold. family equations that arises in analysis the context graph-based learning and contains, as important examples, satisfied by graph Laplacian eigenvectors. particular, prove high probability interior global estimates for solutions Poisson equations. Our results can be used to show eigenvectors are, with probability, essentially regular constants depending explicitly their corresponding eigenvalues. relies probabilistic coupling argument suitable walks at continuum level, an interpolation method extending functions point clouds As byproduct our general results, obtain $L^\infty$ approximate $\mathcal{C}^{0,1}$ convergence rates toward eigenfunctions weighted Laplace--Beltrami operators. scale like $L^2$ established J. Calder N. García Trillos, arXiv:1910.13476, 2019.