In the paper An Impossibility Theorem for Clustering, Jon Kleinberg introduced an axiomatic framework for clustering and showed that his set of axioms are inconsistent. One of the axioms is the Richness Axiom: any partition of the input can be produced by some distance function.
My question is:
Isn't the richness axiom unnatural?
The axiom implies that both trivial partitions: the one which puts all elements in the same cluster, and the one in which each cluster is a singleton, are both considered as valid clusterings. However, this is counter-intuitive in real life situations. In many (i guess most) real life situations we wouldn't have both these clusterings considered as valid simultaneously. so i guess the richness axiom could be modified. do we truly need all partitions to be considered as valid clusterings?