## 2 Theoretical Background: Deep Learning, Sparse Auto-Encoder (SAE) and Support Vector Machine (SVM)  
Deep learning may be faced as a special class of artificial neural networks (ANN), normally with multiple layers. This characteristic allows the possibility to firmly adjust ranked sets of network parameters that make this machine learning technique capable of obtaining concise information from raw data. These sets of parameters may be applied to resolve problems like prognosis or pattern identification, for example. Among the available deep learning strategies, one can highlight Long Short-Term Memory [37], Convolutional Neural Network [38] and Auto-Encoder [30]. This last-mentioned strategy is the focus of the present paper, as previously noted. The main feature of the ANN called Auto-Encoder (AE) is its type of operation: The architecture of an AE network is constructed to achieve an output that approximates its input. A simple AE with three layers is presented in Figure 1 to clarify this statement, which also introduces important elements, namely:  
* Function \(f\): represents the code function that operates over the input signal vector \(\mathbf{x}\) (a single series of dynamic measurements, for instance). * Vector \(\mathbf{h}\): stands for the result of a function \(f\) over a signal vector \(\mathbf{x}\). In other words, it is the feature vector. * Function \(g\): represents the code function that operates over the feature vector \(\mathbf{h}\). This function reconstructs an approximation of the original input signal vector \(\mathbf{x}\). * Vector \(\mathbf{y}\): stands for the result of a function \(g\) over the feature vector \(\mathbf{h}\). In other words, it is the reconstruction of the auto-encoder's input. The code function \(f\) applied to the input data \(\mathbf{x}\) leads to the learned features \(\mathbf{h}=f(\mathbf{x})\). On the other hand, the decode function \(g\) applied to \(\mathbf{h}\) leads to output \(\mathbf{y}=g(\mathbf{h})\). The principal idea is to write a cost function to calculate the AE parameters that produce \(\mathbf{y}\approx\mathbf{x}\). At this point, it is important to observe that the learning process of an AE is unsupervised. There are three possibilities for AE concerning the dimensionality of the \(\mathbf{h}\) and \(\mathbf{x}\) vector, \(K\) and \(M\), respectively: For \(K>M\), the complexity of \(\mathbf{h}\) is higher than the one of \(\mathbf{x}\) vector. This is not desirable if one wants to \"learn\" essential features of \(\mathbf{x}\). For \(K=M\), this second hypothesis may be faced as a transformation of \(\mathbf{x}\), which also does not reduce the complexity of the problem. AEs placed on the third possibility (\(K<M\)) are called Undercomplete Auto Encoders, and they are often used for data mapping problems, like those addressed in the present work. The reduction of data's dimension allows identifying significant parameters of \(\mathbf{x}\) that are appropriate for identifying structural damage (\(\mathbf{h}\) vector). Instead of purely minimizing the cost function \(Z(\mathbf{x},g(f(\mathbf{x})))\), which calculates the difference between \(\mathbf{x}\) and \(\mathbf{y}=g(f(\mathbf{x}))\), a sparse penalty \(T(f(\mathbf{x}))\) may be added in the function \(Z\) during the training phase of the network. Function \(T(f(\mathbf{x}))\) is selected to increase the capacity of the AE to represent \(\mathbf{x}\) with a small number of \(\mathbf{h}\) components. In this case, the AE is called a Sparse Auto-Encoder (SAE). [PERSON] et al. (2016) presented an excellent overview of AE and SAE, which is recommended for further inquiries. To illustrate the above definitions, a typical SAE network is presented in Figure 2. It has an equal number of encoding and decoding layers (two in this figure) and a central layer from which the internal codes \(\mathbf{h}\) are derived. The symmetric configuration is mandatory for SAE networks.