Monotone linkage clustering and quasi-concave set functions

Abstract

Greedily seriating objects one by one is implicitly employed in many heuristic clustering procedures which can be described in terms of a linkage function measuring entity-to-set dissimilarities. A well-known clustering technique, single linkage clustering, can be considered as an example of the seriation procedures (based actually on the minimum spanning tree construction) leading to the global maximum of a corresponding “minimum split” set function. The purpose of this work is to extend this property to the wide class of so-called monotone linkages. It is shown that the minimum split functions of monotone linkages can be maximized greedily. Moreover, this class of set functions is proven to coincide with the class of so-called quasi-concave set functions.