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Graph Laplacians and their Convergence on Random Neighborhood Graphs

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons83958

Hein,  M
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons76237

von Luxburg,  U
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Hein, M., Audibert, J.-Y., & von Luxburg, U. (2007). Graph Laplacians and their Convergence on Random Neighborhood Graphs. Journal of Machine Learning Research, 8, 1325-1370. Retrieved from http://jmlr.csail.mit.edu/papers/volume8/hein07a/hein07a.pdf.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-CD4D-F
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.