PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Graph Laplacians and their convergence on random neighborhood graphs
Matthias Hein, Jean-Yves Audibert and Ulrike Von Luxburg
Journal of Machine Learning Research Volume 8, pp. 1325-1368, 2007.

Abstract

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.

EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Learning/Statistics & Optimisation
Theory & Algorithms
ID Code:3149
Deposited By:Jean-Yves Audibert
Deposited On:29 December 2007