  Kernel $k$-means clustering is a powerful tool for unsupervised learning of
non-linearly separable data. Since the earliest attempts, researchers have
noted that such algorithms often become trapped by local minima arising from
non-convexity of the underlying objective function. In this paper, we
generalize recent results leveraging a general family of means to combat
sub-optimal local solutions to the kernel and multi-kernel settings. Called
Kernel Power $k$-Means, our algorithm makes use of majorization-minimization
(MM) to better solve this non-convex problem. We show the method implicitly
performs annealing in kernel feature space while retaining efficient,
closed-form updates, and we rigorously characterize its convergence properties
both from computational and statistical points of view. In particular, we
characterize the large sample behavior of the proposed method by establishing
strong consistency guarantees. Its merits are thoroughly validated on a suite
of simulated datasets and real data benchmarks that feature non-linear and
multi-view separation.
