An Algorithm for the Separation-Preserving Transition of Clusterings

TL;DR

This paper introduces an algorithm that enables a smooth, step-by-step transition between two separable clusterings while maintaining cluster separability, leveraging linear programming and polytope theory.

Contribution

The paper presents a novel algorithm that uses polytope walks and sensitivity analysis to achieve a gradual, separability-preserving transition between clusterings.

Findings

Algorithm successfully transitions between clusterings while preserving separability.

Utilizes polytope boundary properties and linear programming techniques.

Provides a theoretical framework for continuous clustering transformations.

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 where 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, 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 re-optimizing the cluster assignment of items.