k-quantiles: L1 distance clustering under a sum constraint

This paper considers centroid-based clustering under ℓ1 distance in which both data points and cluster centers are subject to a sum constraint on their components. A closed-form solution is derived for the cluster center optimization problem, enabling an interpretation as a sample quantile of the cluster. An adaptive sampling initialization step is also adopted to provide a guarantee on expected clustering cost as well as empirical improvements. Experiments on synthetic data indicate that the advantages of the proposed algorithms increase as clusters become more concentrated and as the dimension increases. An application to clustering employee job role profiles highlights the utility of ℓ1 distance in promoting sparse, interpretable cluster centers.