# A Study of OWA Operators Learned in Convolutional Neural Networks
## 2 Preliminaries
### Convolutional Neural Networks  
Generally speaking, Neural Networks (NNs) [25] are algorithms developed for pattern recognition. They are built as an aggregation of multiple simpler units, neurons, arranged in a network. These neurons, when properly trained, can specialize in different simple tasks, letting the whole system learn complex behaviors.  
One of the key features of these systems is the training algorithm, based in the idea of backpropagation [26]. This algorithm evaluates the network over an input, and then computes the errors between the obtained and expected outputs. These errors are then backpropagated through the network, from output to input, updating all the parameters along the way to increase the system accuracy. This technique enables the training of extremely complex NNs over large datasets.  
Different network architectures have been designed for specific purposes. Initially, the Multilayer Perceptron (MLP) [27] was proposed as a multi-purpose neural network. MLPs are dense networks that arrange the neurons in layers, each one fully connected to all the nodes in the next and previous layers. This leads to complex networks with many weights to train. Some architectures try to reduce this complexity according to the constraints of specific problems. Convolutional Neural Networks (CNNs) [11; 12] are another well-known variant of NNs where convolutional filters are combined with pooling layers to deal with multidimensional data such as image classification.  
There are two main layers that define CNNs, convolutional and pooling layers. Convolutional layers determine the number of convolutions to apply, each one with its independent weights (the convolutional filter). Since each convolution aggregates the input into a single channel image, also known as feature map, the convolutional layer as a whole generates a new image with as many channels as filters are applied in the layer. In these convolutional layers, most of the information is aggregated across channels, while vertically and horizontally the information is aggregated only on the immediate neighborhood of each pixel. This means that the information codified in a single pixel of a feature map comes only from its spatial neighborhood and ignores the rest of the image.  
Pooling layers are used to aggregate the information in the spatial dimensions of the image [24]. These layers keep the channels of the image intact and reduce the dimensionality of the image in the vertical and horizontal dimensions. To perform this reduction, blocks of a certain width and height are defined independently for each channel, and the information in those blocks is aggregated to a single element. This aggregation is usually performed using operators such as the maximum or the average, although there has been some work also on the usage of general OWAs [18; 19].  
In CNNs, by design, the network tends to operate by exploiting local information in the convolutional layers early in the architecture, and aggregating it into global scale information on the later stages of the network, after each application of a pooling layer. The OWA layer studied in this work tries to change this architecture, letting the network learn some global information in the early layers. Examples of this type of architecture specific to the image classification problem are LeNet [28], the VGG family of models [29], ResNet [30], and DenseNet [31], among others.