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  Graph Laplacians and their Convergence on Random Neighborhood Graphs

Hein, M., Audibert, J.-Y., & von Luxburg, U. (2007). Graph Laplacians and their Convergence on Random Neighborhood Graphs. The Journal of Machine Learning Research, 8, 1325-1370.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-CD4D-F Version Permalink: http://hdl.handle.net/21.11116/0000-0003-CB25-F
Genre: Journal Article

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 Creators:
Hein, M1, 2, Author              
Audibert, J-Y, Author
von Luxburg, U1, 2, Author              
Affiliations:
1Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society, ou_1497795              
2Max Planck Institute for Biological Cybernetics, Max Planck Society, Spemannstrasse 38, 72076 Tübingen, DE, ou_1497794              

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 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.

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 Dates: 2007-06
 Publication Status: Published in print
 Pages: -
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 Table of Contents: -
 Rev. Type: -
 Identifiers: BibTex Citekey: 4613
 Degree: -

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Title: The Journal of Machine Learning Research
Source Genre: Journal
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Publ. Info: Cambridge, MA : MIT Press
Pages: - Volume / Issue: 8 Sequence Number: - Start / End Page: 1325 - 1370 Identifier: ISSN: 1532-4435
CoNE: https://pure.mpg.de/cone/journals/resource/111002212682020_1