Data driven estimation of Laplace-Beltrami operator

TL;DR

This paper introduces a data-driven method for selecting bandwidth parameters in graph Laplacian approximations of Laplace-Beltrami operators on manifolds, improving their practical applicability in data analysis.

Contribution

It establishes an oracle inequality for the unnormalized graph Laplacian, enabling a theoretically justified bandwidth selection procedure based on Lepski's method.

Findings

Provides a rigorous framework for bandwidth tuning in graph Laplacians.

Enables more accurate approximation of Laplace-Beltrami operators from data.

Facilitates improved manifold learning and data analysis techniques.

Abstract

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.