On the Consistency of Metric and Non-Metric K-medoids

Abstract

We establish the consistency of K-medoids in the context of metric spaces. We start by proving that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution under general conditions, including a wide selection of loss functions. This asymptotic equivalence, in turn, enables us to apply the work of Pärna (1986) on the consistency of K-means. This general approach applies also to non-metric settings where only an ordering of the dissimilarities is available. We consider two types of ordinal information: one where all quadruple comparisons are available; and one where only triple comparisons are available. We provide some numerical experiments to illustrate our theory.We establish the consistency of K-medoids by first showing that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution, and then leveraging the work of Pärna (1986) on the consistency of K-means in metric spaces.