Towards a Theoretical Foundation for Laplacian-Based Manifold Methods

In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifold-motivated” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and p...

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Hauptverfasser:	Belkin, Mikhail, Niyogi, Partha
Format:	Tagungsbericht
Sprache:	eng
Schlagworte:	Applied sciences Artificial intelligence Computer science control theory systems Exact sciences and technology Geodesic Distance Heat Equation Heat Kernel Learning and adaptive systems Point Cloud Spectral Cluster
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creator	Belkin, Mikhail Niyogi, Partha
description	In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifold-motivated” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacian-based manifold methods. We show that under certain conditions the graph Laplacian of a point cloud converges to the Laplace-Beltrami operator on the underlying manifold. Theorem 1 contains the first result showing convergence of a random graph Laplacian to manifold Laplacian in the machine learning context.
doi_str_mv	10.1007/11503415_33
format	Conference Proceeding
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subjects	Applied sciences Artificial intelligence Computer science control theory systems Exact sciences and technology Geodesic Distance Heat Equation Heat Kernel Learning and adaptive systems Point Cloud Spectral Cluster
title	Towards a Theoretical Foundation for Laplacian-Based Manifold Methods
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