Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

Abstract

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold $${\mathcal {M}}$$ in $$\mathbb {R}^d$$ as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $$O\Big (\big (\frac{\log n}{n}\big )^\frac{1}{2m}\Big )$$ to the eigenvalues and eigenfunctions of the weighted Laplace–Beltrami operator of $${\mathcal {M}}$$ . No information on the submanifold $${\mathcal {M}}$$ is needed in the construction of the graph or the “out-of-sample extension” of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in $$\mathbb {R}^d$$ in infinity transportation distance.