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Consistency of Spectral Clustering

2008

Article

ei


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.

Author(s): von Luxburg, U. and Belkin, M. and Bousquet, O.
Journal: Annals of Statistics
Volume: 36
Number (issue): 2
Pages: 555-586
Year: 2008
Month: April
Day: 0

Department(s): Empirical Inference
Bibtex Type: Article (article)

Digital: 0
DOI: 10.1214/009053607000000640
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

Links: PDF

BibTex

@article{4403,
  title = {Consistency of Spectral Clustering},
  author = {von Luxburg, U. and Belkin, M. and Bousquet, O.},
  journal = {Annals of Statistics},
  volume = {36},
  number = {2},
  pages = {555-586},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = apr,
  year = {2008},
  month_numeric = {4}
}