An Analysis of the Convergence of Graph Laplacians

An Analysis of the Convergence of Graph Laplacians

Report Number
792
Authors
Daniel Ting, Ling Huang, and Michael Jordan
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.

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