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

#### Autoren: Nicolas Garcia Trillos, Moritz Gerlach, Matthias Hein, Dejan Slepcev (2019)

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a -dimensional submanifold in as the sample size increases and the neighborhood size tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of .
No information on the submanifold 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 in infinity transportation distance.

Zeitschrift:
Foundations of Computational Mathematics

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