  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.
