# A Study of OWA Operators Learned in Convolutional Neural Networks
## 3 Related Work  
OWA operators have been widely used since they were first proposed by [PERSON] [2], thanks to their ability to represent and generalize some well-known aggregations and linguistic operators, such as the minimum, the maximum or the median, among others. Many ways of determining their specific weighting vectors have been developed in the literature [6; 7; 8]. Most of these methods are supervised approaches, focused on translating an expert opinion to a set of weights [6]. These systems for weight determination are often related to specific families of parametric OWAs, where the expressivity of the operator is reduced to adapt to some usual property constraints [7]. Some of these parameterizations [8; 22; 23] were presented in Section 2.2. In other problems, where more expressivity is needed, some works have explored ways to determine the weights of the operators from data [9]. In this work, we will focus on this second approach, but integrating the operators in a standard NN backpropagation loop, designed for complex problems with large numbers of data. The combination of OWA operators and NN has also been already explored in the literature [14; 15; 16; 17; 18; 19; 20; 21]. Usually, the OWA operator (or other fuzzy measure-based aggregation function) is introduced outside of the classifier [15; 16; 17; 21], for example, to aggregate an ensemble of classifiers, but some other approaches have also been explored [14; 18; 19; 20]. One of the most ambitious proposals of integration methods is the Linear Order Statistic Neuron developed by [PERSON] et al. [14]. In this work, the authors propose an artificial neuron fully based on an OWA operator, modifying the original simple perceptron idea. When assembled into a complete network, these kinds of neurons would probably exhibit very different properties to those of the NNs, although it is a preliminary work. The most common approach, and arguably the most natural application of OWA operators (and other families of aggregation functions), is the aggregation of the output of multiple classifiers, which can include Neural Networks [15; 16; 17; 21]. In these schemes, each classifier of the ensemble is trained independently from the rest, and the operator is used to aggregate the outputs into a single one, improving the final system accuracy at the expense of increasing the global complexity. Another approach that has been explored is the integration of a Fuzzy Measure-based operator (of which the OWA operator is a particular case) in the pooling layer of a CNN [18; 19]. In this case, the aggregation used to reduce the dimensionality of a feature map in a pooling layer is replaced with other types of aggregations. Specifically, in these works, the authors employed a Choquet-like integral and show an improvement in results compared to the usual mean and max aggregations. Finally, the main inspiration for this work has been the \"Fuzzy Layer\" proposed in [20]. In [20], the fuzzy layer is composed of several OWA operators that are applied to the feature maps generated by a convolutional layer, thus generating a set of new fused maps. These feature maps serve as the input to the next layer, completely replacing the original ones. Regarding the selection of the operators, they manually established fixed operators. In contrast, our OWA layer proposal in [13] is also composed of several OWA operators that generate new fused feature maps, but with important differences. With respect to the usage of the new feature maps, in our proposal, the fused feature maps are used to augment the information of the network, preserving the original feature maps. Moreover, we do not fix beforehand the weights of the OWA operators, but instead, we let the network learn them as parameters. This process of automatic weight determination is the focus of this work. The OWA layer that we explore in this work was previously defined in [13; 38], where we presented some preliminary results about the impact of the layer in CNNs.