The paper investigates whether the three axioms of scale invariance, richness, and consistency—known to be mutually exclusive for flat clustering—can be jointly satisfied by hierarchical clustering. It demonstrates that, unlike the flat case, there exist uncountably many hierarchical clustering methods that meet all three axioms, termed admissible methods. The authors construct several such methods, explore a refinement partial order among them, and show that while the set of admissible methods is diverse, every admissible method shares a common backbone of well‑separated clusters.
By Daichi Kuroda, Maximilien Dreveton, Matthias Grossglauser, Patrick Thiran
arXiv:2604. 23628v2 Announce Type: replace-cross Abstract: Hierarchical clustering is a fundamental task in data analysis, but classical methods have long lacked a principled objective function.
By Ryuki Tsukuba, Kazutoshi Ando
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 $\mathcal{F}$.
arXiv:2607. 13217v1 Announce Type: cross Abstract: 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.
By Micha{\l} Szyfelbein, Dariusz Dereniowski
arXiv:2606. 18972v1 Announce Type: cross Abstract: Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem.
By Connor Simpson, Ricardo J. G. B. Campello
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.
The paper introduces the Universal Clustering Problem (UCP), a framework that captures the optimisation core common to many clustering methods by maximizing a polynomial‑time computable partition utility over a finite metric space. It proves UCP is NP‑hard through reductions from graph colouring and exact cover by 3‑sets, showing that popular algorithms such as k‑means, GMMs, DBSCAN, spectral clustering, and affinity propagation inherit this intractability. The authors argue that this unified hardness explains typical failure modes—like local optima and greedy merge traps—and suggest moving toward stability‑aware objectives and interaction‑driven formulations with explicit guarantees.
By Angshul Majumdar
The paper investigates hierarchical clustering under an individual fairness constraint that limits relative distortion within local k‑nearest neighborhoods. It formulates this as a feasibility problem over dominated ultrametrics, characterizes the minimal multiplicative slack needed, identifies a sharp local threshold, proves stability under bounded perturbations, establishes monotonicity in k, and demonstrates a Θ(log n) separation between local and global realizability. Experiments on synthetic and real‑world datasets corroborate the theoretical findings.
By Binita Maity, Shrutimoy Das
The paper introduces absolute cluster indices that assess both compactness and separability of clusters, moving beyond relative measures commonly used in clustering validation. It defines a compactness function for each cluster and a set of neighboring points for cluster pairs to evaluate cluster quality and overall distribution margin. These indices are applied to determine the true number of clusters and are compared against widely-used validity indices on synthetic and real-world datasets.
By Adil M. Bagirov, Ramiz M. Aliguliyev, Nargiz Sultanova, Sona Taheri
arXiv:2411. 01576v3 Announce Type: replace Abstract: The explainable clustering problem was first posed by Moshkovitz et al.
By Maximilian Fleissner, Maedeh Zarvandi, Debarghya Ghoshdastidar
arXiv:2607. 10305v1 Announce Type: cross Abstract: Byzantine-robust aggregation rules such as multi-Krum assume a central coordinator, and decentralising them is obstructed by the rules themselves: they are globally coupled, non-associative, and discontinuous, so an ulpscale perturbation can flip the selected subset, moving the output by a non-vanishing amount.
By Ryan Gillespie
arXiv:2409. 00743v4 Announce Type: replace-cross Abstract: In recent years, much of the research on clustering algorithms has primarily focused on enhancing their accuracy and efficiency, frequently at the expense of interpretability.
By Lianyu Hu, Mudi Jiang, Junjie Dong, Xinying Liu, Zengyou He