# Constructing Clustering Transformations

## TL;DR

This paper introduces methods for transforming one clustering into another using linear programming and network theory, providing a new metric for clustering differences and bounds on the related partition polytopes.

## Contribution

It presents a novel approach for clustering transformation based on a clustering-difference graph and elementary moves, along with bounds on the circuit diameter of partition polytopes.

## Key findings

- Developed a clustering-difference graph model for transformations.
- Provided methods for decomposing transformations into elementary moves.
- Established bounds on the circuit diameter of partition polytopes.

## Abstract

Clustering is one of the fundamental tasks in data analytics and machine learning. In many situations, different clusterings of the same data set become relevant. For example, different algorithms for the same clustering task may return dramatically different solutions. We are interested in applications in which one clustering has to be transformed into another; e.g., when a gradual transition from an old solution to a new one is required. In this paper, we devise methods for constructing such a transition based on linear programming and network theory. We use a so-called clustering-difference graph to model the desired transformation and provide methods for decomposing the graph into a sequence of elementary moves that accomplishes the transformation. These moves are equivalent to the edge directions, or circuits, of the underlying partition polytopes. Therefore, in addition to a conceptually new metric for measuring the distance between clusterings, we provide new bounds on the circuit diameter of these partition polytopes.