Clustering via matrix exponentiation

Abstract

Given a set of n points with a matrix of pairwise similarity measures, one would like to partition the points into clusters so that similar points are together and different ones apart. We present an algorithm requiring only matrix exponentiation that performs well in practice and bears an elegant interpretation in terms of random walks on a graph. Under a certain mixture model involving planting a partition via randomized rounding of tailored matrix entries, the algorithm can be proven effective for only a single squaring. It is shown that the clustering performance of the algorithm degrades with larger values of the exponent, thus revealing that a single squaring is optimal.