An objective function for order preserving hierarchical clustering

• URL: http://arxiv.org/abs/2109.04266v4
• Date: Tue, 10 Dec 2024 18:31:50 GMT
• Title: An objective function for order preserving hierarchical clustering
• Authors: Daniel Bakkelund,
• Abstract summary: We present a theory and an objective function for similarity-based hierarchical clustering of probabilistic partial orders. 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 $[x]le'[y]$.
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• 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.

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