Gradient Based Clustering

TL;DR

This paper introduces a versatile, gradient-based clustering method that is computationally efficient, applicable to various cost functions including non-Bregman types, and converges reliably to meaningful cluster configurations.

Contribution

It presents a general, gradient-based iterative clustering algorithm with simple updates, applicable to diverse cost functions, and provides convergence analysis and empirical validation.

Findings

Algorithm converges to fixed points from arbitrary initializations.

Effective on real datasets, including non-Bregman clustering.

Applicable to a broad class of clustering costs, including Huber loss.

Abstract

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.