# A Soft-Voting Ensemble Classifier for Detecting Patients Affected by COVID-19
## 2 Materials and Methods
### Inception Networks  
CNNs are a class of deep neural networks widely used for images analysis. Basically, a CNN implements a stack of several modules that perform convolutions, introduce non-linearity to the model, downsample the features and classify the images. These operations make use of convolutional, ReLU, pooling, and fully connected layers. The convolutional layer represents the core building block of a CNN. It applies a set of convolution filters to the input feature map, producing an output feature map. Optimal values for the filters are learned by the CNN during the training phase. The ReLU layer is used to apply the rectifier function, with the aim of increasing the model's non-linearity after convolution. The pooling layer is used to downsample the features by implementing a dimensionality reduction of the feature map. Finally, the fully connected layer performs the classification task using the features extracted by the previous layers. Inception networks represented a milestone in CNNs. Before the advent of Inception, the design of CNN architectures was focused on increasing the convolutional layers. However, networks designed according to this principle have some limitations. In particular, the number of parameters of the network increases with the number of network layers. As a consequence, the parameters might not fit in memory. Moreover, models built on very deep networks can be affected by overfitting, especially when working with not large datasets. To deal with these limitations, Inception moved from fully connected network architectures to sparsely connected network architectures. This idea resulted in the so-called Inception module. The naive Inception module consists of a pooling path and filters of different sizes (i.e., \(1\times 1\), \(3\times 3\), \(5\times 5\)) which perform convolutions on the input image (see Figure 1a). The output generated by these operators is concatenated into a single output vector representing the input of the next layer. Some convolutions can become prohibitively expensive on top of a convolutional layer with a large number of filters. To deal with this issue, the Inception module has been modified by adding \(1\times 1\) convolutions to compute reductions before the expensive \(3\times 3\) and \(5\times 5\) convolutions (see Figure 1b). This Inception module has been used to build the so-called Inception-V1 network [22] consisting of 27 layers. Inception-V2 and Inception-V3 are two incremental optimizations presented in the same paper [23]. Architectural changes implemented in Inception-V2 are aimed at reducing the computational cost with respect to Inception-V1. This was obtained implementing smart factorization methods aimed at making convolutions computationally more efficient. In Inception-V2, convolution filters with kernel size \(3\times 3\) were factorized to two \(5\times 5\) convolutions. This change affects the performance of the network in terms of computational time and accuracy. Notably, a \(5\times 5\) convolution is 2.78 times more expensive than a \(3\times 3\) convolution. Moreover, using \(3\times 3\) is preferable to larger kernel size, which may affect the accuracy of a neural network due to the representational bottleneck. Additionally, \(n\times n\) convolutions were factorized to a stack of \(1\times n\) and \(n\times 1\) convolutions (see Figure 2). Converting \(3\times 3\) convolutions into \(1\times 3\) and \(3\times 1\) convolutions was 33% cheaper in terms of computational cost. Inception-V3 implements all changes reported for Inception-V2 and additionally makes use of (i) RMSprop optimizer, (ii) label smoothing regularization, (iii) \(7\times 7\) factorized convolution, and (iv) batch normalization in the fully connected layer. Experiments carried out in the paper presenting both architectures showed that better performances were obtained using Inception-V3 also outperforming Inception-V1. Figure 1: Inception-V1 modules.