Towards a Theoretical Foundation for Laplacian-Based Manifold Methods

• Published in: Learning Theory pp. 486 - 500
• Main Authors: Belkin, Mikhail, Niyogi, Partha
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• 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; Laplace operator; Digitizing; Differential operator; Artificial intelligence; Laplacian; Random graph
• ISBN: 3540265562; 9783540265566
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11503415_33

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.