Kernel k-Groups via Hartigan's Method

Energy statistics was proposed by Sz\'{e}kely in the 80's inspired by Newton's gravitational potential in classical mechanics, and it provides a model-free hypothesis test for equality of distributions. In its original form, energy statistics was formulated in Euclidean spaces. More recently, it was generalized to metric spaces of negative type. In this paper, we consider a formulation for the clustering problem using a weighted version of energy statistics in spaces of negative type. We show that this approach leads to a quadratically constrained quadratic program in the associated kernel space, establishing connections with graph partitioning problems and kernel methods in unsupervised machine learning. To find local solutions of such an optimization problem, we propose an extension of Hartigan's method to kernel spaces. Our method has the same computational cost as kernel k-means algorithm, which is based on Lloyd's heuristic, but our numerical results show an improved performance, especially in high dimensions.