Spectral clustering with more than K eigenvectors

Spectral clustering techniques are heuristic algorithms aiming to find approximate solutions to difficult graph-cutting problems, usually NP-complete, which are useful to clustering. A fundamental working hypothesis of these techniques is that the optimal partition of K classes can be obtained from the first K eigenvectors of the graph normalized Laplacian matrix L_N if the gap between the K-th and the K+1-th eigenvalue of L_N is sufficiently large. If the gap is small a perturbation may swap the corresponding eigenvectors and the results can be very different from the optimal ones.In this paper we suggest a weaker working hypothesis: the optimal partition of K classes can be obtained from a K-dimensional subspace of the first M>K eigenvectors, where M is a parameter chosen by the user. We show that the validity of this hypothesis can be confirmed by the gap size between the K-th and the M+1-th eigenvalue of L_N. Finally we present and analyse a simple probabilistic algorithm that generalizes current spectral techniques in this extended framework. This algorithm gives results on real world graphs that are close to the state of the art by selecting correct K-dimensional subspaces of the linear span of the first M eigenvectors, robust to small changes of the eigenvalues.