An Algorithm for the Separation-Preserving Transition of Clusterings

Abstract

The separability of clusters is one of the most desired properties in clustering. There is a wide range of settings in which different clusterings of the same data set appear. We are interested in applications for which there is a need for an explicit, gradual transition of one separable clustering into another one. This transition should be a sequence of simple, natural steps that upholds separability of the clusters throughout. We design an algorithm for such a transition. We exploit the intimate connection of separability and linear programming over bounded-shape partition and transportation polytopes: separable clusterings lie on the boundary of partition polytopes and form a subset of the vertices of the corresponding transportation polytopes, and circuits of both polytopes are readily interpreted as sequential or cyclical exchanges of items between clusters. This allows for a natural approach to achieve the desired transition through a combination of two walks: an edge walk between two so-called radial clusterings in a transportation polytope, computed through an adaptation of classical tools of sensitivity analysis and parametric programming, and a walk from a separable clustering to a corresponding radial clustering, computed through a tailored, iterative routine updating cluster sizes and reoptimizing the cluster assignment of items. Funding: Borgwardt gratefully acknowledges support of this work through National Science Foundation [Grant 2006183] Circuit Walks in Optimization, Algorithmic Foundations, Division of Computing and Communication Foundations; through Air Force Office of Scientific Research [Grant FA9550-21-1-0233] The Hirsch Conjecture for Totally-Unimodular Polyhedra; and through Simons Collaboration [Grant 524210] Polyhedral Theory in Data Analytics. Happach has been supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research.