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Consistency of spectral clustering. (English) Zbl 1133.62045
Summary: 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. 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.

MSC:
62H30Classification and discrimination; cluster analysis (statistics)
05C90Applications of graph theory
65C60Computational problems in statistics
62G20Nonparametric asymptotic efficiency
05C50Graphs and linear algebra
47N30Applications of operator theory in probability theory and statistics
62G20Nonparametric asymptotic efficiency