Graph Laplacians and their Convergence on Random Neighborhood Graphs

This paper determines the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero and shows that for a uniform measure on the submanifold all graph LaPLacians have the same limit up to constants.

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.