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 up- date 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.

We thank the reviewers for their useful comments and suggestions. The work of A. Armacki and S. Kar was partially supported by the National Science Foundation under grant CNS-1837607. The work of D. Bajovic and D. Jakovetic is supported by the European Union's Horizon 2020 Research and Innovation program under grant agreements No 957337 and 871518. This paper reflects only the authors' views and the European Commission cannot be held responsible for any use which may be made of the information contained therein.