  We propose a general approach for distance based clustering, using the
gradient of the cost function that measures clustering quality with respect to
cluster assignments and cluster center positions. The approach is an iterative
two step procedure (alternating between cluster assignment and cluster center
updates) and is applicable to a wide range of functions, satisfying some mild
assumptions. The main advantage of the proposed approach is a simple and
computationally cheap update rule. Unlike previous methods that specialize to a
specific formulation of the clustering problem, our approach is applicable to a
wide range of costs, including non-Bregman clustering methods based on the
Huber loss. We analyze the convergence of the proposed algorithm, and show that
it converges to the set of appropriately defined fixed points, under arbitrary
center initialization. In the special case of Bregman cost functions, the
algorithm converges to the set of centroidal Voronoi partitions, which is
consistent with prior works. Numerical experiments on real data demonstrate the
effectiveness of the proposed method.
