Matrix Aggregation Operators

Inmaculada Gutiérrez; Asier Urio-Larrea; J. Tinguaro Rodríguez; Daniel Gómez; Javier Montero; Humberto Bustince

stat.MLSep 23, 2026

Matrix Aggregation Operators

Authors: Inmaculada Gutiérrez, Asier Urio-Larrea, J. Tinguaro Rodríguez, Daniel Gómez, Javier Montero, Humberto Bustince

Organizations: Faculty of Statistical Studies, Complutense University of Madrid · Instituto Universitario de Estadística y Ciencia de Datos, Complutense University of Madrid · Department of Statistics, Computer Science and Mathematics, Universidad Pública de Navarra · Institute of Smart Cities · Institute of Smart Cities (ISC), Universidad Pública de Navarra · Faculty of Mathematics, Complutense University of Madrid · 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.

Figures & tables

	$A_{1}$	$A_{2}$	$\cdots$	$A_{K}$
$x_{1}$	$\mu_{A_{1}}(x_{1})$	$\mu_{A_{2}}(x_{1})$	$\cdots$	$\mu_{A_{K}}(x_{1})$
$x_{2}$	$\mu_{A_{1}}(x_{2})$	$\mu_{A_{2}}(x_{2})$	$\cdots$	$\mu_{A_{K}}(x_{2})$
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$
$x_{N}$	$\mu_{A_{1}}(x_{N})$	$\mu_{A_{2}}(x_{N})$	$\cdots$	$\mu_{A_{K}}(x_{N})$

Table 1 :

u_{ij}=\mu_{A_{j}}(x_{i})

membership degree for

N

objects and

K

classes.

Figure 1 : Relationships between the four basic row and column symmetry properties (within boxes) and the derived symmetry properties, including global symmetry. Arrows should be read as implications, while the symbol

\odot

denotes the conjunction of properties.

Figure 2 : Partitions $\mathcal{P}_{1},\dots,\mathcal{P}_{6}$ of an example 2-dimensional dataset containing $N=1024$ samples balancedly drawn from $K_{true}=4$ isotropic Gaussian distributions. A scaling factor $\sigma_{scale}=0.3$ is used, resulting in a moderate overlap of two of the four groups. The partitions were obtained by means of the FCM algorithm, using fuzziness parameter $m=2$ and Euclidean distance, and assigning each sample to the cluster with maximum membership degree ( 15 ). A black cross identifies each cluster centroid.

Index	$K=1$	$K=2$	$K=3$	$K=4$	$K=5$	$K=6$
$S_{0}(\mathbf{U}_{K}^{FCM})$	1.0000	0.6260	0.5330	0.4765	0.4431	0.4250
$S_{0.5}(\mathbf{U}_{K}^{FCM})$	1.0000	0.8868	0.9283	0.8918	0.8311	0.7837
$S_{1}(\mathbf{U}_{K}^{FCM})$	1.0000	0.9999	0.9999	0.9999	0.9999	0.9990
$S_{0}(\mathbf{U}_{K}^{COV})$	0.2132	0.2724	0.4880	0.5742	0.5742	0.5742
$S_{0.5}(\mathbf{U}_{K}^{COV})$	0.4290	0.6171	0.7795	0.8260	0.8334	0.8432
$S_{1}(\mathbf{U}_{K}^{COV})$	0.7375	0.9833	0.9853	0.9907	0.9889	0.9853

Table 2 : Behavior of the different indices for the sequence of partitions shown in Figure 3 .

Figure 3 : Value of each index for partitions $\mathcal{P}_{K}$ , $K=1,\dots,50$ . The top panel shows the MEGCIs obtained from the $\mathbf{U}_{K}^{FCM}$ matrices of membership values returned by FCM, as well as the corresponding XB index. The bottom panel shows the MEGCIs obtained from the $\mathbf{U}_{K}^{COV}$ matrices of coverage degrees, together with the curvature index corresponding to the cohesion measure $S_{0.5}(\mathbf{U}_{K}^{COV}$ ).

Parameter	Symbol	Values
Number of Groups	$K_{true}$	$\{2,4,8,16,32\}$
Dimensionality	$p$	$\{2,5,10,15\}$
Number of Samples	$N$	$\{1024,2048,4096,8192\}$
Std. Deviation	$\sigma_{scale}$	$0.10,0.15,0.20,0.25,0.30$

Table 3 : Parameters used for the generation of the synthetic datasets.

Methods	Hits	Acc. (%)	Methods	Hits	Acc. (%)
PC	147	36.75	$CV^{0.05}$	217	54.25
XB	154	38.50	$CV^{0.15}$	255	63.75
XBH	162	40.50	$CV^{0.25}$	267	66.75
DB	191	47.75	$CV^{0.35}$	280	70.00
SS	230	57.50	$CV^{0.45}$	293	73.25
CH	249	62.25	$CV^{0.55}$	297	74.25

Table 4 : Performance at the NCE task of baseline and MEGCI-based ICVIs across the 400 synthetic datasets.

Figure 4 : Proportion of correct identifications of the actual number of groups $K$ across the considered 400 datasets, by method.

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A Unified Framework for Gradient Aggregation in Multi-Objective Optimization

Zeou Hu, Kelvin Ho, Yaoliang Yu

1Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada · 3Vector Institute · 2Computer Science, The Chinese University of Hong Kong

Convex Optimization Multi-Objective Optimization Convergence Federated Learning Convex Sets Unified Framework

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Many machine learning problems involve multiple inherent trade-offs that are best addressed by gradient-based multi-objective optimization (MOO) algorithms. Existing methods are often proposed with various motivations, analyzed case by case, and differ algorithmically in how the component gradients are aggregated at each step. In this work, we develop a unifying framework for gradient aggregation in MOO, establishing (optimal) rates of convergence to Pareto stationarity, the standard measure of performance in MOO. Central to our analysis is a sufficient alignment condition, from which we derive a theorem showing that non-conflicting directions, when chosen within the convex hull of gradients, form a fundamental sufficient condition for convergence. We further show that feasibility can be ensured through projection onto the dual cone, broadening the scope of methods that admit convergence guarantees. In parallel, we present a primal optimization perspective of gradient aggregation that encompasses established algorithms, clarifies their theoretical relationships, and enables the design of new variants. As an illustration, we introduce capped MGDA, derived from a CVaR-based formulation, and demonstrate its robustness in adversarial federated learning. Finally, we validate our theory through experiments on synthetic problems and practical benchmarks.

Abstract

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