An Analysis of the Convergence of Graph Laplacians

TL;DR

This paper generalizes the analysis of graph Laplacians by removing smoothness assumptions, includes kNN graphs, introduces a kernel-free framework, and discusses how to achieve desirable spectral properties through graph construction choices.

Contribution

It removes smoothness assumptions in graph Laplacian analysis, extends the framework to kNN graphs, and introduces a kernel-free approach for analyzing various graph constructions.

Findings

Generalized graph Laplacian analysis without smoothness assumptions

Included analysis of kNN graphs and locally linear embedding (LLE)

Provided guidelines for achieving desirable spectral properties

Abstract

Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a well-behaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also introduce a kernel-free framework to analyze graph constructions with shrinking neighborhoods in general and apply it to analyze locally linear embedding (LLE). We also describe how for a given limiting Laplacian operator desirable properties such as a convergent spectrum and sparseness can be achieved choosing the appropriate graph construction.