An objective function for order preserving hierarchical clustering
This work addresses the challenge of maintaining order relations in hierarchical clustering for structured data like DAGs, offering a novel solution with practical improvements.
The paper tackles the problem of hierarchical clustering for probabilistic partial orders and DAGs by developing a theory and objective function to ensure order preservation, and it introduces a polynomial-time approximation algorithm that significantly outperforms existing methods.
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.