The Annals of Statistics

Consistency of spectral clustering

Ulrike von Luxburg, Mikhail Belkin, and Olivier Bousquet

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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.

Article information

Ann. Statist., Volume 36, Number 2 (2008), 555-586.

First available in Project Euclid: 13 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Spectral clustering graph Laplacian consistency convergence of eigenvectors


von Luxburg, Ulrike; Belkin, Mikhail; Bousquet, Olivier. Consistency of spectral clustering. Ann. Statist. 36 (2008), no. 2, 555--586. doi:10.1214/009053607000000640.

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