Open Access
On fuzzy implications derived from general overlap functions and their relation to other classes
(MDPI, 2023) Pinheiro, Jocivania; Santos, Helida; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Santiago, Regivan; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called (GO, N)-implication functions. We also investigate their properties, the aggregation of (GO, N)-implication functions, their characterization and the intersections with other classes of fuzzy implication functions.
Open Access
Generalizing max pooling via (a, b)-grouping functions for convolutional neural networks
(Elsevier, 2023) Rodríguez Martínez, Iosu; Da Cruz Asmus, Tiago; Pereira Dimuro, Graçaliz; Herrera, Francisco; Takáč, Zdenko; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Due to their high adaptability to varied settings and effective optimization algorithm, Convolutional Neural Networks (CNNs) have set the state-of-the-art on image processing jobs for the previous decade. CNNs work in a sequential fashion, alternating between extracting significant features from an input image and aggregating these features locally through ‘‘pooling" functions, in order to produce a more compact representation. Functions like the arithmetic mean or, more typically, the maximum are commonly used to perform this downsampling operation. Despite the fact that many studies have been devoted to the development of alternative pooling algorithms, in practice, ‘‘max-pooling" still equals or exceeds most of these possibilities, and has become the standard for CNN construction. In this paper we focus on the properties that make the maximum such an efficient solution in the context of CNN feature downsampling and propose its replacement by grouping functions, a family of functions that share those desirable properties. In order to adapt these functions to the context of CNNs, we present (𝑎, 𝑏)- grouping functions, an extension of grouping functions to work with real valued data. We present different construction methods for (𝑎, 𝑏)-grouping functions, and demonstrate their empirical applicability for replacing max-pooling by using them to replace the pooling function of many well-known CNN architectures, finding promising results.
Open Access
On generalized overlap and grouping indices in n-dimensional contexts
(Springer, 2025-05-08) Asmus, Tiago da Cruz; Dimuro, Graçaliz Pereira; Lucca, Giancarlo; Marco Detchart, Cedric; Santos, Helida; Camargo, Heloisa A.; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Overlap and grouping indices are functions measuring, respectively, the fuzzy intersection and fuzzy union of two fuzzy sets. They have been applied successfully in several fields, such as in interpolative fuzzy systems, fuzzy rule-based classification systems and comparison of fuzzy inference rules. Overlap and grouping indices can be built employing overlap and grouping functions, respectively, which are possibly non-associative aggregation functions with features that provide good results when applied to practical bivariate problems. Many studies have generalized the concepts of overlap and grouping functions to be applied in n-dimensional problems. However, the concepts of overlap/grouping indices have not been generalized in similar pattern. Since the associative property may not hold, their application in n-dimensional domains, for comparing more than two fuzzy sets at a time, is not immediate, which limit their application in such contexts. The objective of this paper is to introduce the concepts of n-dimensional and general overlap/grouping indices, with special attention to the development of their construction methods based on generalized overlap/grouping functions. As an application example, we introduce the concept of n-dimensional Jaccard index, with a construction method based on n-dimensional overlap/grouping indices, providing an n-dimensional fuzzy set similarity score.

Bustince Sola, Humberto

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