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Erschienen in:
2005 | OriginalPaper | Buchkapitel
Towards a Theoretical Foundation for Laplacian-Based Manifold Methods
verfasst von : Mikhail Belkin, Partha Niyogi
Erschienen in: Learning Theory
Verlag: Springer Berlin Heidelberg
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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. 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.