Data driven estimation of Laplace-Beltrami operator

Part of Advances in Neural Information Processing Systems 29 (NIPS 2016)

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Frederic Chazal, Ilaria Giulini, Bertrand Michel


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.