Open Access
April 2008 Consistency of spectral clustering
Ulrike von Luxburg, Mikhail Belkin, Olivier Bousquet
Ann. Statist. 36(2): 555-586 (April 2008). DOI: 10.1214/009053607000000640


Consistency is a key property of all statistical procedures analyzing randomly sampled data. 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.


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Ulrike von Luxburg. Mikhail Belkin. Olivier Bousquet. "Consistency of spectral clustering." Ann. Statist. 36 (2) 555 - 586, April 2008.


Published: April 2008
First available in Project Euclid: 13 March 2008

zbMATH: 1133.62045
MathSciNet: MR2396807
Digital Object Identifier: 10.1214/009053607000000640

Primary: 62G20
Secondary: 05C50

Keywords: consistency , convergence of eigenvectors , graph Laplacian , spectral clustering

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 2 • April 2008
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