Regularity-based spectral clustering and mapping the Fiedler-carpet

Abstract

We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted k k -means algorithm. In the case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, while in the case of edge-weighted graphs, a normalized Laplacian-based clustering. In the latter case, it is proved that a spectral gap between the ( k − 1 ) \left(k-1) st and k k th smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2 k − 1 {2}^{k-1} , but only the first k − 1 k-1 eigenvectors are used in the representation. The ensemble of these eigenvectors constitute the so-called Fiedler-carpet.