The Annals of Statistics
- Ann. Statist.
- Volume 36, Number 2 (2008), 555-586.
Consistency of spectral clustering
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.
Ann. Statist. Volume 36, Number 2 (2008), 555-586.
First available in Project Euclid: 13 March 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62G20: Asymptotic properties
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
von Luxburg, Ulrike; Belkin, Mikhail; Bousquet, Olivier. Consistency of spectral clustering. Ann. Statist. 36 (2008), no. 2, 555--586. doi:10.1214/009053607000000640. https://projecteuclid.org/euclid.aos/1205420511