Density-aware Hierarchical Clustering Based on Element-Categorized Connection Subgraphs

Clustering is a fundamental data mining technique for pattern recognition through unsupervised learning. Among various clustering methods, hierarchical clustering, density-based clustering, and graph clustering stand out as representative approaches. For hierarchical clustering, it can be categorized into agglomerative and divisive modes to construct clusters in a recursive manner. The key aspect of both modes is the calculation of inter-cluster similarity, which determines whether to merge the sub-clusters into one cluster or divide a current cluster into sub-clusters. Traditionally, the similarity is derived from pairwise distances, often overlooking density variations and structural connectivity in graphs. To address this, we propose a density-aware hierarchical clustering method based on element-categorized connection subgraphs (DHC-ECS), which effectively integrates the hierarchical clustering, density-based clustering, and graph clustering. Particularly, a novel inter-cluster similarity metric is introduced that considers not only distances but also the element categorization in the KNN connection subgraphs, kernel density estimation, and local connectivity within sub-clusters. Extensive evaluations on heterogeneous benchmark datasets demonstrate that DHC-ECS exhibits superior overall performance in terms of clustering accuracy and parameter robustness compared with the baseline methods (including AChameleon, RNN-DBSCAN, McDPC, and G-RMS). The work indicates the great potential of the proposed clustering algorithm for low-dimensional datasets by leveraging local density and graph-structured connectivity (i.e., the duality of vertices and edges), as well as the possibility to determine an intrinsic threshold, reducing the reliance on manual parameter tuning.

Hierarchical $\mathcal{F}$-Clustering: Approximation and Hardness of Clustering into Trees and Bounded Diameter Graphs

Consider the following variation on the Hierarchical Clustering problem: Usually, while building a hierarchical clustering, one recursively partitions the data until each cluster becomes a singleton. We relax the halting condition of the recursive process to stop whenever the remaining cluster is a graph belonging to a class $\mathcal{F}$. We call this problem Hierarchical $\mathcal{F}$-Clustering and we measure the quality of any solution using adapted Dasgupta's clustering objective. We study two natural choices of $\mathcal{F}$: trees and graphs of bounded diameter. We present the first polynomial time $\mathcal{O}(\log n\cdot\log\log n)$ and $\mathcal{O}(\log n)$-approximation algorithms for clustering into trees and bounded diameter graphs respectively. Our main technical contribution is a framework for approximating such problems based on linear programming. In fact, we characterize graphs classes $\mathcal{F}$ for which our approach can be applied and show that it includes both trees and bounded diameter graphs. However, our ideas are not limited to them and might be useful for other structures as well. Broadly speaking, our framework applies whenever the corresponding flat clustering problem, which we call $p_{\mathcal{F}}$-Partitioning, admits a natural ILP formulation together with a rounding procedure with provable approximation guarantees. Intuitively, given a set of vertices called terminals, the problem is to find an edge set whose removal results in satisfying certain vertex-dependent structural predicate for each terminal. We then use these ingredients to build clustering trees with the aforementioned approximation guarantees. To complement these results, we show that both Hierarchical Clustering into trees and into bounded diameter graphs cannot be approximated within any constant factor under the Small Set Expansion Hypothesis.

Hierarchical Partial-Order Models for Ranking

Rank aggregation combines information from ordered lists ranking items by preference. Classical parametric models for such data, including the Mallows and Plackett-Luce models, assume the orders concentrate around one or more complete consensus rankings. Recent work relaxes the total-order assumption by allowing the consensus structure to be a partial order (poset), allowing for incomparabilities in preferences. However, in many applications preference data exhibit group structure. We introduce hierarchical partial order (HPO) models, which extend poset-based models to accommodate grouped data through a hierarchy of latent posets. This framework, which parallels mixture model extensions of the Mallows and Plackett-Luce models, enables principled sharing of information across groups while preserving partial-order structure. We show that the Plackett-Luce model and its hierarchical variants are special cases of HPO-models. We develop a hierarchical clustering extension (HCPO) for unsupervised clustering in settings where group labels are unknown. Bayesian inference for the latent poset hierarchy is performed using Markov chain Monte Carlo methods. Experiments on synthetic and real-world datasets, including pairwise acoustic preference data and LLM agent traces, demonstrate that the proposed HPO and HCPO models outperform existing approaches in both predictive performance and structural interpretability.

FOSC-X: An Extended Framework for Optimal Local Cuts and Non-Horizontal Cluster Selection from Clustering Hierarchies

Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem. Existing approaches focus on finding a single optimal solution. We introduce FOSC-X, a framework for extracting the top-M globally optimal flat clusterings from local, non-horizontal cuts of a hierarchical cluster tree, while optionally enforcing constraints on the number of clusters. This enables automatic identification of multiple high-quality alternative clusterings that capture different aspects of the hierarchical structure. Without constraints, the top-M problem can be solved in polynomial time using dynamic programming, exploiting the property that locally optimal partial candidates within subtrees can be combined to form globally optimal solutions while automatically determining the number of clusters. However, this can lead to solutions with numbers of clusters that are ultimately undesirable -- e.g., too large to be meaningful or practically analysed within a particular application domain. Imposing cluster-count constraints breaks the optimality property underlying the unconstrained dynamic programming approach, since locally optimal partial candidates may no longer combine into feasible globally optimal solutions. FOSC-X addresses this challenge through a dynamic programming strategy that maintains compact sets of feasible candidates using lower and upper feasibility bounds while pruning infeasible or dominated combinations. The resulting method guarantees optimal rankings of the top-M solutions with linear-time complexity in the number of cluster nodes and dataset size, both with and without cluster-count constraints. Experiments show that FOSC-X efficiently reveals alternative clustering structures overlooked by single-solution extraction methods.

Semi-Supervised Hyperbolic Hierarchical Clustering with Set-Level Structural Priors

Semi-supervised hierarchical clustering aims to learn a tree structure consistent with data patterns and user-provided supervision. Supervision is usually given as leaf-level relations, such as pairwise must-link/cannot-link constraints or triplet-wise must-link-before constraints. Although useful for regulating local sample relations, such supervision does not directly indicate which samples should form coherent subtrees. Consequently, the non-leaf structure of the learned tree may deviate from the hierarchical organization preferred by ground-truth labels. To address this limitation, we propose a semi-supervised hyperbolic hierarchical clustering method with set-level structural priors. The main contribution is to introduce sets as basic modeling units for hierarchy learning. Each set denotes samples expected to cohere within a subtree and is induced from leaf-level supervision together with a learned constraint-consistent similarity structure. These sets act as soft structural priors for subtree-level supervision, allowing supervision to guide non-leaf hierarchy formation beyond local leaf-level relations. Specifically, we first learn constraint-consistent embeddings to obtain a reliable set partition, then construct constraint-induced sets and estimate inter-set similarities to form set-level structural priors. Finally, these priors are incorporated into a hyperbolic hierarchy objective for continuous tree optimization. Experiments on eleven benchmark datasets and ablation studies show that the proposed method consistently improves label consistency over representative hierarchical clustering baselines while also enhancing similarity-based tree quality.