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In this paper we study Lipschitz regularity of elliptic PDEs on geometricgraphs, constructed from random data points. The data points are sampled from adistribution supported on a smooth manifold. The family of equations that westudy arises in data analysis in the context of graph-based learning andcontains, as important examples, the equations satisfied by graph Laplacianeigenvectors. In particular, we prove high probability interior and globalLipschitz estimates for solutions of graph Poisson equations. Our results canbe used to show that graph Laplacian eigenvectors are, with high probability,essentially Lipschitz regular with constants depending explicitly on theircorresponding eigenvalues. Our analysis relies on a probabilistic couplingargument of suitable random walks at the continuum level, and an interpolationmethod for extending functions on random point clouds to the continuummanifold. As a byproduct of our general regularity results, we obtain highprobability $L^\infty$ and approximate $\mathcal{C}^{0,1}$ convergence ratesfor the convergence of graph Laplacian eigenvectors towards eigenfunctions ofthe corresponding weighted Laplace-Beltrami operators. The convergence rates weobtain scale like the $L^2$-convergence rates established by two of the authorsin previous work.

Lipschitz regularity of graph Laplacians on random data clouds