Spectral Methods for Automatic Multiscale Data Clustering

Spectral clustering is a simple yet powerful method for finding structure in data using spectral properties of an associated pairwise similarity matrix. This paper provides new insights into how the method works and uses these to derive new algorithms which given the data alone automatically learn different plausible data partitionings. The main theoretical contribution is a generalization of a key result in the field, the multicut lemma [7]. We use this generalization to derive two algorithms. The first uses the eigenvalues of a given affinity matrix to infer the number of clusters in data, and the second combines learning the affinity matrix with inferring the number of clusters. A hierarchical implementation of the algorithms is also derived. The algorithms are theoretically motivated and demonstrated on nontrivial data sets.