Energy Clustering

Energy statistics was proposed by Sz\'{e}kely in the 80's inspired by the Newtonian gravitational potential from classical mechanics, and it provides a hypothesis test for equality of distributions. It was further generalized from Euclidean spaces to metric spaces of strong negative type, and more recently, a connection with reproducing kernel Hilbert spaces (RKHS) was established. Here we consider the clustering problem from an energy statistics theory perspective, providing a precise mathematical formulation yielding a quadratically constrained quadratic program (QCQP) in the associated RKHS, thus establishing the connection with kernel methods. We show that this QCQP is equivalent to kernel $k$-means optimization problem once the kernel is fixed. These results imply a first principles derivation of kernel $k$-means from energy statistics. However, energy statistics fixes a family of standard kernels. Furthermore, we also consider a weighted version of energy statistics, making connection to graph partitioning problems. To find local optimizers of such QCQP we propose an iterative algorithm based on Hartigan's method, which in this case has the same computational cost as kernel $k$-means algorithm, based on Lloyd's heuristic, but usually with better clustering quality. We provide carefully designed numerical experiments showing the superiority of the proposed method compared to kernel $k$-means, spectral clustering, standard $k$-means, and Gaussian mixture models in a variety of settings.