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Matrix Aggregation Operators

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

arXiv:2609.28562v1stat.MLcs.LG

TL;DR

Many applications require aggregating membership degrees arranged across objects and fuzzy sets, but the paper identifies a lack of formal structure for these matrix operators. It formalizes MAOs and studies their properties, including symmetry and decomposability. It shows that some MAOs are not decomposable and that simultaneous row and column symmetry can make matrix structure irrelevant to aggregation.

  • Problem

    Applications use membership-degree matrices, but the paper identifies no formal framework for this class of aggregation operators, while flattening can discard meaningful row-column relationships.

    Source: §1 · p. 3, §3.1 · p. 7

  • Method

    The paper defines MAOs, analyzes decomposability and symmetry, and introduces MEGCIs as a family of MAOs.

    Source: §1 · p. 4, §3.1 · p. 7

  • Results

    Some MAOs are neither row- nor column-decomposable, and simultaneous row and column symmetry implies global symmetry, making matrix structure irrelevant to aggregation.

    Source: §3.2 · p. 9, §4.1 · p. 11, §4.2 · p. 14

  • Takeaways & Limitations

    The paper applies MEGCIs to clustering quality assessment while defining aggregation operators that act directly on matrices.

    Source: §1 · p. 4, §3.1 · p. 7

    The definition requires only non-strict entrywise monotonicity and does not require sensitivity to every matrix entry; restricted or directional monotonicity is left for future research.

    Source: §3.1 · p. 8

Abstract

from arXiv · show

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.

1. Introduction

Many applications aggregate membership degrees across both objects and fuzzy sets, so their information is naturally represented as a matrix. The paper formalizes matrix aggregation operators (MAOs), studies their properties, and introduces MEGCIs for clustering quality assessment.

1. Introduction

  • Matrices represent membership degrees linking multiple objects to multiple fuzzy sets, and aggregation maps this information to a single value.The paper cites decision-making, classification, and overlap or grouping indices as application settings.Source: §1 · pp. 3–4, Table 1 · p. 4
  • Flattening a matrix can discard meaningful relationships between its rows and columns, while sequential row- or column-wise aggregation covers only decomposable operators.The paper notes that some MAOs, including a generalization of the polarization index [12], cannot be expressed in decomposable form.Source: §1 · p. 4
  • The paper formalizes MAOs, analyzes their decomposability and symmetry, and introduces maximum entropy global coverage indices (MEGCIs) for clustering quality assessment.The paper illustrates MEGCI applicability to clustering through a computational study.Source: §1 · p. 4

2. Preliminaries

The preliminaries establish vector aggregation operators and the grouping and ordered weighted averaging operators used in the paper. They also introduce maximum-entropy OWA weights as a way to obtain an OWA operator for a specified orness degree.

2. Preliminaries

  • An aggregation operator maps [0, 1]^n to [0, 1] and satisfies boundary conditions and coordinatewise monotonicity; symmetry is an additional property requiring invariance to input permutations.Symmetry is not part of the stated basic aggregation-operator conditions.Source: §2 · p. 5
  • An n-dimensional grouping function is symmetric, continuous, and non-decreasing, reaching 0 only when all inputs are 0 and 1 when any input is 1.Grouping functions extend the disjunctive dual of conjunction-oriented overlap functions to multiple fuzzy degrees.Source: §2 · p. 5
  • An OWA operator applies a weighting vector to the ordered inputs, spanning aggregation behaviors from minimum to maximum.Its symmetry follows from sorting the data before applying the weights; orness measures closeness to maximum-like versus minimum-like behavior.Source: §2 · p. 6
  • For a specified orness degree d, maximum-entropy weights minimize Shannon entropy subject to that degree and yield a MEOWA operator.For d in (0, 1), the weights are positive and distribute importance across all aggregated objects as uniformly as possible for that orness.Source: §2 · p. 6
  • The operators are defined on [0, 1], though the paper notes that another real interval may also be used.The stated alternative is any interval I contained in R.Source: §2 · p. 6

3. Matrix Aggregation Operators and Decomposability

The paper defines MAOs as matrix-to-scalar aggregation operators and distinguishes them from sequential row- or column-wise constructions. It shows that decomposability is a proper subclass: some MAOs admit neither form, and row- and column-decomposability need not coincide.

3.1. Definition of Matrix Aggregation Operator

  • A matrix aggregation operator maps an N × K matrix of [0, 1] entries to [0, 1], satisfying zero and one boundary conditions and entrywise monotonicity.The paper relates each fixed matrix flattening to a corresponding vector aggregation operator; the flattened operator depends on the chosen mapping.Source: §3.1 · p. 7
  • Trace is a MAO, and aggregating selected matrix entries with a vector aggregation operator also gives a MAO; the determinant is given as a matrix operator that fails the MAO conditions.Trace uses the diagonal entries, so it disregards off-diagonal entries; the broader issue of restricted monotonicity and directional monotonicity is left for future research.Source: §3.1 · p. 8

3.2. Decomposability of Matrix Aggregation Operators

  • Row- or column-decomposable MAOs first aggregate each row or column, then aggregate the resulting vector; aggregation operators in the stages suffice to construct either form.The paper defines a MAO as decomposable if it is row-decomposable or column-decomposable, and notes that trace is both.Source: §3.2 · pp. 8–9
  • Some MAOs are not decomposable: the 2 × 2 operator F1 is neither row- nor column-decomposable, while separate examples show each form need not imply the other.F1 is identified as a particular instance of the general polarization index proposed in.Source: §3.2 · p. 9

4. Symmetry of Matrix Aggregation Operators

The section defines global, row, and column symmetry for matrix aggregation operators, then establishes implications among these properties. It shows that some combinations make matrix structure irrelevant, while other MAOs retain narrower symmetry patterns.

4.1. Global Symmetry

  • Global symmetry is equivalent to symmetry of the unique flattened aggregation operator, so flattening preserves the result only in this case.For a globally symmetric MAO, the choice of flattening does not affect the resulting vector aggregation operator.Source: §4.1 · p. 11

4.2. Row and Column Symmetry

  • Internal column symmetry implies external row symmetry, while internal row symmetry implies external column symmetry.Source: §4.2 · p. 13
  • A MAO is globally symmetric exactly when it satisfies both internal row symmetry (IRS) and internal column symmetry (ICS), making arbitrary entry rearrangement immaterial.In that case it can be treated as a symmetric vector aggregation operator, with the matrix’s row and column meanings lost.Source: §4.2 · p. 14
  • Total row symmetry and total column symmetry each imply external row-and-column symmetry.Source: §4.2 · p. 14
  • Transposition exchanges external row symmetry with external column symmetry, and internal row symmetry with internal column symmetry.Source: §4.2 · p. 15
  • Examples include MAOs that satisfy some row or column symmetries without global symmetry; a non-decomposable MAO can also satisfy external row-and-column symmetry.The trace operator has diagonal symmetry despite satisfying none of the eight studied properties.Source: §4.2 · pp. 15–16

5. Matrix Aggregation Operators in applications

The section recasts established procedures from decision-making, classification, overlap and grouping analysis, and clustering as matrix aggregation operators. These examples show how matrix structure and the meanings of rows and columns shape the aggregation used.

Applications

  • Across these applications, rows and columns carry different semantic meanings, and MAOs provide a framework for aggregating while preserving that two-dimensional structure.Source: §5 · pp. 16, 19

Example 5 (Multicriteria Decision Making Problems)

  • Decision procedures aggregate criterion or decision-maker evaluations into values for alternatives, with Max–Min and Min–Max strategies satisfying TRS but not GS.The Min–Max strategy is framed for risk or cost; fixed criterion weights yield only ERS, while MaxiMax is globally symmetric.Source: §5 · p. 17

Example 7 (Fuzzy Rule-Based Classification Systems)

  • In fuzzy rule-based classification, class certainty aggregates association degrees by column, then selects the class with maximum certainty; max aggregation is globally symmetric, while mean aggregation verifies TCS.Source: §5 · p. 18

Example 8 (Grouping and Overlap Indices)

  • Overlap and grouping indices transform membership degrees for multiple fuzzy sets over a universe into a single value and can be viewed as MAOs.Source: §5 · p. 18

Example 9 (Fuzzy Clustering)

  • Fuzzy clustering validity indices aggregate the membership matrix into a value reflecting partition quality, including proxy notions of cluster compactness and separability.The Xie–Beni index is given as an example.Source: §5 · p. 18

Example 10 (Polarization Measures)

  • The JDJ polarization index genuinely combines information across rows and columns and cannot be decomposed into successive row and column aggregations.Source: §5 · pp. 18–19

6. Maximum Entropy Global Coverage Indexes

The section introduces maximum entropy global coverage indices (MEGCIs), built by applying a grouping function to matrix rows and aggregating the results with maximum-entropy OWA weights. Their orness parameter orders each family between limiting minimum- and maximum-based indices, with a coverage interpretation for clustering.

Maximum Entropy Global Coverage Indexes

  • A GCI is totally row symmetric, equals 1 exactly when every row contains a 1, and equals 0 exactly when every matrix entry is 0.Source: §6 · p. 19
  • OWA-based GCIs can weight rows according to their grouping values, allowing lower-grouping rows to receive more weight and making the index more demanding.The requirement level can be quantified by the weights’ orness degree.Source: §6 · p. 20
  • Maximum-entropy weights provide a unique positive weighting vector for each interior orness degree and distribute importance as uniformly as possible across rows.Source: §6 · p. 20
  • For each grouping function G and orness degree d in (0, 1), a unique MEGCI is obtained by combining G with the corresponding maximum-entropy OWA operator.The construction is an instance of the result that composing a grouping function with an OWA operator having strictly positive weights yields a GCI.Source: §6 · pp. 20–21
  • Within each grouping-function family, increasing orness orders the MEGCIs between limiting operators min∘G and max∘G, which can serve as computable endpoints.Source: §6 · p. 21
  • With row-wise maximum grouping, MEGCIs form clustering-quality indices: low orness emphasizes the least-covered object, while high orness emphasizes the best-covered object.At d = 0.5, the index is described as the most balanced; the limits are min_i(max_j(u_ij)) and max_i,j(u_ij).Source: §6 · p. 21

7. Application: MEGCI-based Cluster Quality Assessment

The section applies MAOs to cluster-quality assessment by aggregating object–cluster association matrices. It contrasts FCM membership degrees with absolute coverage degrees and shows how coverage-based MEGCIs can support cohesion measures and internal validity indexes.

7. Application: MEGCI-based Cluster Quality Assessment

  • MEGCI-based indexes assess partition quality by aggregating an object-by-cluster association matrix through a matrix aggregation operator.The matrix records associations between N objects and K clusters; deriving a partition index from it is an aggregation task.Source: §7 · pp. 21–22

7.1. MEGCI-based Internal Cluster Cohesion Measures and Cluster Validity Indexes

  • Only when matrix entries are coverage degrees can certain MEGCIs be interpreted as cluster cohesion or compactness measures.The section also describes transforming these cohesion measures into internal cluster validity indexes to assess partition quality.Source: §7 · p. 22
  • The illustration uses standardized two-dimensional data with four true Gaussian groups and examines FCM partitions for cluster counts up to 50.The first six partitions are shown; two of the four groups have moderate overlap.Source: §7.1 · p. 22, Figure 2 · p. 23
  • The example compares FCM membership degrees, which sum to one per object, with coverage degrees that decay exponentially with distance and need not sum to one.The coverage values are intended to express absolute rather than relative object–cluster association.Source: §7.1 · pp. 22–23

K and UCOV

The operators use maximum grouping to associate each object with its most-associated cluster, then summarize membership or coverage across objects.

K and UCOV

  • The grouping function G = max assigns each object to the cluster with which it is most associated.The resulting S_d indexes summarize the minimum, mean, or maximum membership or coverage across objects and their assigned clusters.Source: K and UCOV · pp. 23–24
  • S1 is excessively stable and non-informative because it considers only the object closest to any centroid.That object's distance tends to be near zero, so the index tends to be approximately 1 for any K.Source: K and UCOV · p. 24
  • Operators with d approximately 1 exhibit the same insensitive behavior, making them unsuitable as cohesion measures.Table 2 summarizes index behavior for the partition sequence shown in Figure 3.Source: K and UCOV · p. 24, Table 2 · p. 24

K ) and S1(UCOV

The passage characterizes operators with d near 1 as insensitive for cohesion measurement.

K ) and S1(UCOV

  • Operators with d approximately 1 are insensitive, rendering them unsuitable as cohesion measures.The passage says the index values tend to be approximately 1 for any K.Source: K ) and S1(UCOV · p. 24

K ) and S0.5(UF CM

Across 400 synthetic datasets, a broad range of MEGCI-based curvature indexes outperforms all tested baseline indexes for estimating cluster count. In the illustrative example, coverage-based cohesion measures behave more usefully than FCM-membership-based measures, while the broader study shows performance depends on the orness setting.

K ) and S0.5(UF CM

  • The curvature method converts monotonic coverage-based MEGCI sequences into an internal validity index by selecting the cluster count that maximizes its curvature ratio.The study computes CV_d for K from 2 through Kmax − 1 and varies orness degree d.Source: K ) and S0.5(UF CM · p. 25
  • In the illustrative dataset, CV 0.5 peaks at K = 4, whereas the XB index reaches its minimum at K = 3.The dataset has four actual groups, including two with moderate overlap.Source: K ) and S0.5(UF CM · p. 26

7.2.1. Experimental Setup

  • The evaluation spans 400 synthetic isotropic-Gaussian datasets and compares KM- and FCM-derived partitions with traditional internal validity indexes.For each algorithm and dataset, partitions are generated for K = 1,…,50; CV_d is tested at 13 orness values.Source: §7.2.1 · p. 26, §7.2 · p. 26

7.2.2. Experimental results

  • MEGCI-based CV_d indexes with d∈[0.15, 0.75] outperform every baseline, with accuracy from 62.50% to 74.25% and a peak at CV 0.55.Only KM-partition CV_d results are reported because they consistently outperformed FCM-partition results in accuracy.Source: Table 4 · p. 27, §7.2.2 · p. 27, Figure 4 · p. 27
  • Mean accuracy favors more conjunctive settings: d∈[0, 0.45] averages 60.46% Acc., compared with 49.63% Acc. for d∈[0.55, 1].This differs from the single best-performing setting, CV 0.55.Source: §7.2.2 · p. 28
  • The authors conclude that coverage-based MEGCI cohesion measures and validity indexes are flexible and useful for assessing clustering quality, including optimal cluster count.Source: §7.2.2 · p. 28

8. Conclusions and further research

The paper formalizes matrix aggregation operators (MAOs), showing that some cannot be decomposed into row- or column-wise aggregation and that matrix structure yields distinct symmetry properties. It introduces maximum entropy global coverage indices (MEGCIs) and evaluates their use in clustering quality assessment.

8. Conclusions and further research

  • MAOs act directly on matrices, preserving their two-dimensional structure; explicit examples, including the polarization index [12], show that some cannot be reduced to independent row- or column-wise processing.Decomposable MAOs form a proper subclass of all MAOs.Source: §8 · p. 28
  • Seven non-equivalent row- and column-symmetry conditions are organized by their logical relationships; requiring the strongest conditions for both dimensions reduces a MAO to an ordinary vector aggregation operator.That simultaneous requirement erases the role of matrix structure.Source: §8 · pp. 28–29
  • MEGCI-based curvature indexes outperformed representative classical internal cluster validity indexes across a broad range of orness degrees on 400 synthetic datasets for estimating cluster number.The study also found meaningful cohesion measures with coverage-based association degrees, but less informative behavior with standard fuzzy c-means memberships.Source: §8 · p. 29
  • MEGCIs combine grouping-based row aggregation with maximum-entropy OWA weights fixed for each orness degree, yielding cluster cohesion measures from coverage-based association degrees.The measures were less informative when applied to standard fuzzy c-means memberships.Source: §8 · p. 29
  • Future work includes extending decomposability and symmetry analyses to properties such as associativity and idempotency, exploring non-decomposable MAOs in image processing or network analysis, and generalizing MAOs to tensor aggregation.Source: §8 · p. 29

Appendix A. Appendix. Proof of existence of non-decomposable MAOs

The appendix proves that a specified 2 × 2 matrix aggregation operator is neither row- nor column-decomposable. Its row-decomposability assumption leads to incompatible requirements under monotonicity.

Appendix A. Appendix. Proof of existence of non-decomposable MAOs

  • Assuming row-decomposability represents the operator using row aggregators followed by a combining aggregator, and imposes contradictory output requirements on selected inputs.The proof names the row aggregators R1 and R2 and the combining aggregator C.Source: Appendix A · p. 33
  • The required values of c and d must be incomparable, but real numbers are totally ordered, so the operator cannot be row-decomposable.Source: Appendix A · p. 33
  • A similar argument rules out column-decomposability, establishing the operator as a non-decomposable MAO.Source: Appendix A · p. 33
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