Graph Laplacians and their convergence on random neighborhood graphs

TL;DR

This paper analyzes the convergence of different graph Laplacians constructed from samples on a submanifold, revealing that only the random walk Laplacian converges to the weighted Laplace-Beltrami operator under non-uniform measures.

Contribution

It provides the first detailed pointwise convergence analysis of three common graph Laplacians on submanifolds, clarifying their limiting behavior.

Findings

All three Laplacians have the same limit under uniform measures.

Only the random walk Laplacian converges to the weighted Laplace-Beltrami operator for non-uniform measures.

The convergence depends on the measure's uniformity and the choice of Laplacian.

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.