An objective function for order preserving hierarchical clustering

TL;DR

This paper introduces a new objective function and theory for order-preserving hierarchical clustering of DAGs, providing a formal framework, an approximation algorithm, and empirical improvements over existing methods.

Contribution

It develops a formal theory and an objective function for order-preserving hierarchical clustering, along with a polynomial-time approximation algorithm.

Findings

The proposed method outperforms existing order-preserving clustering techniques.

The objective function effectively balances order relations and similarity.

The approach provides a formal classification of order-preserving dendrograms.

Abstract

We present a theory and an objective function for similarity-based hierarchical clustering of probabilistic partial orders and directed acyclic graphs (DAGs). Specifically, given elements $x \le y$ in the partial order, and their respective clusters $[x]$ and $[y]$, the theory yields an order relation $\le'$ on the clusters such that $[x]\le'[y]$. The theory provides a concise definition of order-preserving hierarchical clustering, and offers a classification theorem identifying the order-preserving trees (dendrograms). To determine the optimal order-preserving trees, we develop an objective function that frames the problem as a bi-objective optimisation, aiming to satisfy both the order relation and the similarity measure. We prove that the optimal trees under the objective are both order-preserving and exhibit high-quality hierarchical clustering. Since finding an optimal solution is NP-hard, we introduce a polynomial-time approximation algorithm and demonstrate that the method outperforms existing methods for order-preserving hierarchical clustering by a significant margin.