Monotone Linkage Clustering and Quasi-Convex Set Functions

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 (actually, based 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 a wide class of the so-called monotone linkages. It is shown that the minimumsplit functions of the monotone linkages can be greedily maximized. Moreover, this class of set functions is proven to coincide with the class of so-called quasi-convex set functions.