Journal
JOURNAL OF COMPUTER AND SYSTEM SCIENCES
Volume 74, Issue 8, Pages 1289-1308Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcss.2007.08.006
Keywords
Laplace-Beltrami operator; Graph Laplacian; Manifold methods
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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. (C) 2007 Published by Elsevier Inc.
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