Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator

Nicolas Garcia Trillos; Moritz Gerlach; Matthias Hein; Dejan Slepcev

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arXiv · 1801.10108

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator

Nicolas Garcia Trillos·Moritz Gerlach·Matthias Hein·Dejan Slepcev

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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 $M$ in $R^d$ 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 $M$. No information on the submanifold $M$ 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 $R^d$ in infinity transportation distance.

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Nicolas Garcia Trillos, Moritz Gerlach, Matthias Hein, Dejan Slepcev. 2018-01-30. Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator. https://arxiv.org/abs/1801.10108

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