The paper introduces the concept of matrix aggregation operators (MAOs), a formal framework for aggregating data naturally arranged in matrices, such as membership degrees in fuzzy systems. It examines properties like decomposability and symmetry, showing that some MAOs cannot be expressed in decomposable form. The authors also propose a new family of MAOs called maximum entropy global coverage indices (MEGCIs), constructed from grouping functions and MEOWA operators, and demonstrate their effectiveness in assessing cluster quality through extensive experiments.
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.
arXiv:2608. 19234v1 Announce Type: new Abstract: Decision-making in complex systems often involves dealing with imprecise or uncertain information, frequently represented using fuzzy sets, particularly Triangular Fuzzy Numbers (TFNs).
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.
arXiv:2607. 23243v1 Announce Type: new Abstract: Fuzzy Integral (FI) based aggregation provides a powerful mechanism for nuanced aggregation, for example, in ensemble approaches or decision-level fusion more generally.
arXiv:2111. 15255v2 Announce Type: replace-cross Abstract: The probabilistic linguistic term has been proposed to deal with probability distributions in provided linguistic evaluations.
The paper investigates whether the impossibility results for flat clustering—specifically Kleinberg’s axioms of scale invariance, richness, and consistency—extend to hierarchical clustering. It demonstrates that, unlike the flat case, there exist uncountably many hierarchical clustering methods that satisfy 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 method shares a common backbone of well‑separated clusters.
arXiv:2608.29045v1 Announce Type: new Abstract: Biclustering, or co-clustering, aims to discover coherent submatrices by grouping rows and columns of a data matrix simultaneously. This local two-dime...
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.
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.
arXiv:2608. 14951v1 Announce Type: new Abstract: Low-rank matrix decompositions can uncover patterns and structure in data and have a number of different applications across many disciplines.
arXiv:2511. 03000v2 Announce Type: replace-cross Abstract: Comparing clusterings is central to evaluating unsupervised models, yet the many existing similarity measures can produce widely divergent, sometimes contradictory, evaluations.
arXiv:2601. 11626v2 Announce Type: replace-cross Abstract: Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD).