Towards a Theoretical Foundation for Laplacian-Based Manifold Methods

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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 practice for a class of Laplacian-based manifold methods. These methods utilize the graph Laplacian associated to a data set for a variety of applications in semi-supervised learning, clustering, data representation. We show that under certain conditions the graph Laplacian of a point cloud of data samples converges to the Laplace-Beltrami operator on the underlying manifold. Theorem 3.1 contains the first result showing convergence of a random graph Laplacian to the manifold Laplacian in the context of machine learning. Key words: Laplace-Beltrami operator, graph Laplacian, manifold methods
Mikhail Belkin, Partha Niyogi
Added 26 Jun 2010
Updated 26 Jun 2010
Type Conference
Year 2005
Where COLT
Authors Mikhail Belkin, Partha Niyogi
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