Title: Kernel k-Groups via Hartigan's Method,
Abstract: Energy statistics was proposed by Székely 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.