# A Study of OWA Operators Learned in Convolutional Neural Networks
## 2 Preliminaries
### Aggregation Functions and OWA Operators  
The main mathematical tool to deal with information fusion is the concept of aggregation function. **Definition 1** (Aggregation function [1; 32]). : _A mapping \(f\colon[0,1]^{N}\to[0,1]\) is called an aggregation function if it satisfies boundary conditions, i.e., \(f(0,\ldots,0)=0\) and \(f(1,\ldots,1)=1\), and increasing monotonicity, i.e., if \(x_{i}\leq y_{i}\) for all \(i\in\{1,\ldots,N\}\), then \(f(x_{1},\ldots,x_{N})\leq f(y_{1},\ldots,y_{N})\)._  
Aggregation functions encompass many different type of functions, such as t-norms and t-conorms, among others. In this work, however, we focus on a specific type, called averaging functions. **Definition 2** (Averaging function [33]). : _An aggregation function \(f\colon[0,1]^{N}\to[0,1]\) is called averaging if \(\min(x_{1},\ldots,x_{n})\leq f(x_{1},\ldots,x_{n})\leq\max(x_{1},\ldots,x_{n})\) for every \(x_{1},\ldots,x_{n}\in[0,1]^{n}\)._  
**Remark 1**. : _Notice that, although averaging functions are defined on \([0,1]\), their domain can be extended to any subinterval of the real line._  
Special cases of averaging functions are the arithmetic mean, the median, the geometric mean, and the OWA operators, which are the basis of this work. OWA operators were first proposed by [PERSON] [2; 3] as a weighted averaging function in which the weights are not associated to a particular input, but to a specific order position based on the input magnitudes. This allows for easy modeling of linguistic terms, especially in the context of linguistic information fusion [4]. **Definition 3** (OWA operator [2; 3]). : _An OWA operator based on the weighting vector \(W=[w_{1},\ldots,w_{n}]\), with the condition \(w_{i}\in[0,1]\) for every \(i=1,\ldots,n\) and \(\sum_{i=1}^{n}=1\), is a mapping \(F\colon\mathcal{R}^{n}\to\mathcal{R}\) such that_  
\[(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}w_{i}x_{\sigma(i)}\,, \tag{1}\]  
_where \(\sigma\colon\{1,\ldots,n\}\to\{1,\ldots,n\}\) is a permutation such that \(x_{\sigma(1)}\geq\cdots\geq x_{\sigma(n)}\)._  
Some notable examples of OWA operators are \(\mathtt{max}\) (\(W_{max}=[1,0,\ldots,0]\)), \(\mathtt{min}\) (\(W_{min}=[0,\ldots,0,1]\)), and the \(\mathtt{arithmetic\ mean}\) (\(W_{avg}=\left[\frac{1}{n},\ldots,\frac{1}{n}\right]\)). In OWA operators, in contrast with the weighted arithmetic mean, the weights are associated with the magnitude of the inputs. However, there are situations in which the weights need to be associated with certain alternative criteria. In order to deal with these situations, an extension of the OWA operator was also introduced, the Induced Ordered Weighted Averaging operators (IOWA) [34]. The IOWA is a general class of OWA operators in which the ordering of the arguments is induced by another variable, called the order-inducing variable. The previous definition of the OWA operator holds, but defining \(\sigma\colon\{1,\ldots,n\}\to\{1,\ldots,n\}\) as a permutation such that \(x^{\prime}_{\sigma(1)}\geq\cdots\geq x^{\prime}_{\sigma(n)}\), based on a set of induced variables \(x^{\prime}_{1},\ldots,x^{\prime}_{n}\) that are, in general, different from \(x_{1},\ldots,x_{n}\). In this work, we study the OWA operators inside an OWA layer (presented later in Section 4), where the objective is the aggregation of 2D feature maps, which do not have an inherent linear order structure. This forces us to define several metrics that map the information in a feature map into a single element, which will then induce the order for the OWA operator. Thus, the underlying function for the OWA layer is in fact an IOWA, although we refer to it as OWA for the sake of simplicity. Observe that the OWA operator allows us to define a wide family of aggregation functions varying from a pure OR operator (maximum) to an AND operator (minimum). In order to characterize the behavior of an OWA operator, [PERSON] introduced two measures [2], called orness and dispersion.