Data driven estimation of Laplace-Beltrami operator
This addresses a theoretical and practical bottleneck in data analysis and machine learning for researchers and practitioners using graph-based methods, though it appears incremental as it builds on existing Lepski's method.
The paper tackles the problem of tuning bandwidth parameters for the unnormalized graph Laplacian approximation of Laplace-Beltrami operators on manifolds, establishing an oracle inequality that enables a data-driven selection procedure.
Approximations of Laplace-Beltrami operators on manifolds through graph Lapla-cians 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 unnormalized 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 [LM15] on the so-called Lepski's method.