An Analysis of the Convergence of Graph Laplacians

July, 2010
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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