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Data driven estimation of Laplace-Beltrami operator

Frederic Chazal · Ilaria Giulini · Bertrand Michel

Area 5+6+7+8 #27

Keywords: [ Nonlinear Dimension Reduction and Manifold Learning ] [ Graph-based Learning ] [ (Other) Statistics ] [ Spectral Methods ]


Approximations of Laplace-Beltrami operators on manifolds through graph Laplacians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unormalized graph Laplacian by establishing an oracle inequality that opens the door to a well-founded data-driven procedure for the bandwidth selection. Our approach relies on recent results by Lacour and Massart (2015) on the so-called Lepski's method.

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