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Journal Article

Consistency of Spectral Clustering

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

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Bousquet,  O
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

von Luxburg, U., Belkin, M., & Bousquet, O. (2008). Consistency of Spectral Clustering. The Annals of Statistics, 36(2), 555-586. doi:10.1214/009053607000000640.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-C9B7-0
Abstract
Consistency is a key property of statistical algorithms when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of the popular family of spectral clustering algorithms, which clusters the data with the help of eigenvectors of graph Laplacian matrices. We develop new methods to establish that for increasing sample size, those eigenvectors converge to the eigenvectors of certain limit operators. As a result we can prove that one of the two major classes of spectral clustering (normalized clustering) converges under very general conditions, while the other (unnormalized clustering) is only consistent under strong additional assumptions, which are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering.