# A Study of OWA Operators Learned in Convolutional Neural Networks
###### Abstract  
Ordered Weighted Averaging (OWA) operators have been integrated in Convolutional Neural Networks (CNNs) for image classification through the OWA layer. This layer lets the CNN integrate global information about the image in the early stages, where most CNN architectures only allow for the exploitation of local information. As a side effect of this integration, the OWA layer becomes a practical method for the determination of OWA operator weights, which is usually a difficult task that complicates the integration of these operators in other fields. In this paper, we explore the weights learned for the OWA operators inside the OWA layer, characterizing them through their basic properties of orness and dispersion. We also compare them to some families of OWA operators, namely the Binomial OWA operator, the Stancu OWA operator and the exponential RIM OWA operator, finding examples that are currently impossible to generalize through these parameterizations. OWA operator; aggregation functions; orness; convolutional neural network; deep learning +
Footnote †: 10: 10.111.10.10.  
are Neural Networks (NN), and in particular Deep Learning (DL), which comprises the training of large NNs over large sets of unstructured data. One of the most successful types of NN used in Deep Learning are Convolutional Neural Networks (CNNs) [11; 12]. CNNs are Neural Networks designed for the processing of spatial data, such as images, where any piece of the input information is strongly correlated to other nearby pieces, which form the neighborhood. This fact is usually exploited by CNNs by using convolutional operations that impose local connectivity constraints on the network weights. This results in layers of processing that only exploit local information, ignoring the global perspective of the data, which only gets considered in the final layers of the network. Aiming at integrating OWA operators in CNNs to improve the performance of CNNs in image classification problems, in a previous work [13], we introduced the OWA layer. The OWA layer modifies the usual CNN architecture by using OWA operators capable of aggregating the feature maps of a CNN based on global information (feature map metrics), even when inserted in the early layers of the network. This modification tries to improve the neighborhood analysis performed by the early convolutional layers of the network by leveraging contextual information about the whole image. Furthermore, the specific implementation proposed is designed to only provide new information without replacing the information already present, minimizing the possibility of any negative impact in the network expressivity. The proposed layer learns its weights at the same time as the rest of the parameters in the CNN, unlike previous proposals of integration of OWA operators into NNs [14; 15; 16; 17; 18; 19; 20], where the OWA operators are considered independently from the network. These previous methods used predetermined weights or independent training methods for the parameters of the NN and the OWA weighting vector [15; 16; 17; 21]. This difference makes the analysis of the learned OWA operators especially interesting and lets us test whether usually employed parameterized families of OWAs are expressive enough, or if directly learning the operator weights results in wider range of OWAs. In particular, we compare the learned operators with three families of OWA operators, the Binomial OWA [8], the Stancu OWA [22], and the exponential RIM OWA [23]. Therefore, in this work, we elaborate on our previous work focusing on the study of the OWA operators learned with the OWA layer and compare their properties to other classical OWA operator families. To perform this analysis, we focus on the orness and dispersion profiles of the learned operators, two measures introduced by [PERSON] [2] that characterize OWA operators in their closeness to the _max_ operator and the concentration of the weights.