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Abstract: Consistency is a key property of all statistical procedures analyzingrandomly sampled data. Surprisingly, despite decades of work, little is knownabout consistency of most clustering algorithms. In this paper we investigateconsistency of the popular family of spectral clustering algorithms, whichclusters the data with the help of eigenvectors of graph Laplacian matrices. Wedevelop new methods to establish that, for increasing sample size, thoseeigenvectors converge to the eigenvectors of certain limit operators. As aresult, we can prove that one of the two major classes of spectral clusteringnormalized clustering converges under very general conditions, while theother unnormalized clustering is only consistent under strong additionalassumptions, which are not always satisfied in real data. We conclude that ouranalysis provides strong evidence for the superiority of normalized spectralclustering.

Author: Ulrike von Luxburg, Mikhail Belkin, Olivier Bousquet



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