On Probabilistic k-Richness of the k-Means Algorithms

With Kleinberg's axiomatic system for clustering, a discussion has been initiated, what kind of properties clustering algorithms should have and have not. As Ackerman et al. pointed out, the static properties studied by Kleinberg and other are not appropriate for clustering algorithms with elements of randomness. Therefore they introduced the property of probabilistic k-richness and claimed, without a proof that the versions of k-means both with random initialisation and k-means++ initialization have this property. We prove that k-means++ has the property of probabilistic k-richness, while k-means with random initialisation for well separated clusters does not. To characterize the latter, we introduce the notion of weak probabilistic k-richness and prove it for this algorithm. For completeness, we provide with a constructive proof that the theoretical k-means has the (deterministic) k-richness property.