Matrix Aggregation Operators

arXiv:2609.28562v1 Announce Type: cross Abstract: Aggregation theory has traditionally focused on operators defined over vectors. However, many applications-including Multi-Criteria Decision Making, Group Decision Making, Fuzzy Rule-Based Classification Systems, and overlap/grouping indices-require aggregating information naturally structured as a matrix of membership degrees (e.g., where a set of objects interacts with a family of fuzzy sets). Despite this, no formal framework has been proposed for this class of operators, partly due to the common practice of flattening matrices into vectors (which discards structural information) and partly due to a reliance on decomposable operators that aggregate rows and columns sequentially. This paper addresses this gap by formalizing the notion of a matrix aggregation operator (MAO). We analyze the decomposability and symmetry properties of MAOs, showing that certain operators cannot be expressed in decomposable form and examining several notions of symmetry. Finally, we introduce a family of MAOs termed maximum entropy global coverage indices (MEGCIs), provide a construction method for them based on combining grouping functions and MEOWA operators, and illustrate their usefulness in cluster quality assessment through an extensive computational study.

Authors:Inmaculada Gutiérrez (1 and 2), Asier Urio-Larrea (3 and 4), J.Tinguaro Rodríguez (5 and 6), Daniel Gómez (1 and 2), Javier Montero (5 and 6), Humberto Bustince (3 and 4) ((1) Faculty of Statistical Studies, Complutense University of Madrid (2) Instituto Universitario de Estadística y Ciencia de Datos, Complutense University of Madrid (3) Department of Statistics, Computer Science and Mathematics, Universidad Pública de Navarra (4) Institute of Smart Cities (ISC), Universidad Pública de Navarra (5) Faculty of Mathematics, Complutense University of Madrid (6) Instituto de Matemática Interdisciplinar, Complutense University of Madrid)

Abstract:Aggregation theory has traditionally focused on operators defined over vectors. However, many applications-including Multi-Criteria Decision Making, Group Decision Making, Fuzzy Rule-Based Classification Systems, and overlap/grouping indices-require aggregating information naturally structured as a matrix of membership degrees (e.g., where a set of objects interacts with a family of fuzzy sets). Despite this, no formal framework has been proposed for this class of operators, partly due to the common practice of flattening matrices into vectors (which discards structural information) and partly due to a reliance on decomposable operators that aggregate rows and columns sequentially. This paper addresses this gap by formalizing the notion of a matrix aggregation operator (MAO). We analyze the decomposability and symmetry properties of MAOs, showing that certain operators cannot be expressed in decomposable form and examining several notions of symmetry. Finally, we introduce a family of MAOs termed maximum entropy global coverage indices (MEGCIs), provide a construction method for them based on combining grouping functions and MEOWA operators, and illustrate their usefulness in cluster quality assessment through an extensive computational study.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2609.28562 [stat.ML]

(or arXiv:2609.28562v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.28562 arXiv-issued DOI via DataCite (pending registration)

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Matrix Aggregation Operators

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