## 2 Materials and Methods
### Deep Neural Network and Autoencoders Model
#### 2.4.1 The Basic Sparse Autoencoder (SAE) Network  
Suppose that \(X=(x(1),x(2),\ldots,x(N))^{T}\) is the set of unlabeled initial face image features for training, where \(x(k)\in R^{d_{x}}\), \(N\) and \(d_{x}\) is denoted as the number of pixels in the images, and the number of facial features is denoted as \(N\). Then, the \(l\)-layer high-level learning features are computed using Equation (8) with \(k\)th features. During the computation \(d_{h}\), hidden number units and current layer \(l\) are utilized. \[h^{l}(k)=\left(h_{1}^{l}(k),h_{2}^{l}(k),\ldots,h_{d_{h}}^{l}(k)\right)^{T} \tag{8}\]  
Here, hidden neurons and units are defined using the superscript and subscripts. The 1 st hidden layer of the \(i\)th unit is denoted as \(h_{i}^{(1)}\) in Figure 8. Here, hidden layer \(l\) processes the \(x\) and \(h^{(l)}\) number of features to identify the input image-related output value. In addition, the sparse autoencoder neural model is shown in Figure 9. The encoder has \(x\) inputs in the input layer and \(h\) hidden layer in the encoder that computes the outputs. Then, the decoder processes the input in \(h\) hidden layer to find the output. During this process, optimal parameters are utilized to reduce the deviations between the outcomes. The variations are reduced to minimize the output reconstruction. Therefore, the sparse autoencoder (SAE) is computed using Equation (9) [27]. \[\mathfrak{L}_{SAE}(\theta)=\left[\frac{1}{N}\sum\
olimits_{k=1}^{N}\left(L \left(x(k),d_{\hat{\theta}}\big{(}e_{\widehat{\theta}}(x(k))\big{)}\right) \right)\right]+\left[\alpha\sum\
olimits_{j=1}^{n}KL(\rho||\hat{\rho}_{j}) \right]+\left[\beta||W||_{2}^{2}\right] \tag{9}\]  
In Equation (9), the sum of the mean square error (SMSE) of the idiom that defines the contradiction among incoming \(x(k)\) and rebuilding \(\hat{x}(k)\) is the overhead of the entire set of data. Furthermore, \(e_{\widehat{\theta}}(\cdot)\) maps incoming \(x\in R^{d_{x}}\) to the hidden illustration \(h\in R^{h_{x}}\), which is computed via \(h=e_{\widehat{\theta}}(x)=s(Wx+b_{h})\), where \(b_{h}\in R^{d_{h}}\) in which bias \(b_{h}\) and \(W\) is a weights of the \(d_{h}\times d_{x}\) matrix. The encoder is represented as \(\widehat{\theta}=(W,b_{h})\) while decoder \(d_{\hat{\theta}}(\cdot)\) plots outcome hidden illustrations \(h\) back into the reconstruction space \(\hat{x}\). \(\hat{x}=d_{\hat{\theta}}=s\big{(}W^{T}h+b_{x}\big{)}\), where \(b_{x}\in R^{d_{x}}\) is defined as bias and \(W^{T}\) is a \(d_{x}\times d_{h}\) denoted as a weight matrix. \(s(\cdot)\) is signified as an activation function; here, logistic sigmoid as \(s(z)=\frac{1}{1+e^{-z}}\), was utilized as an activation function for neuron \(z\). Therefore, the decoder is parameterized by \(\theta=\big{(}W^{T},b_{x}\big{)}\). The transposition of the matrix of weights \(W\) results in the matrix of weights \(W^{T}\) of the inverse designation. The Autoencoder successfully minimizes the weight matrix to half its size. The pre-activation of the output layers of the autoencoder, \(\theta=(W,b_{h},b_{x})\), may be written as \(y=W^{T}s(Wx+b_{h})+b_{x}\) using three parameters. Therefore, the rebuilding of the decoder, \(\hat{x}\), can be determined using \(\hat{x}=s(y)\). The Autoencoder training aims to minimize the reconstruction error mentioned in the first phrase while optimizing the parameters \(\theta=(W,b_{h},b_{x})\). The difference between the  
Figure 9: The simple sparse autoencoder architecture. incoming \(x\) and the reconstruction \(\hat{x}\) made by the decoder \(d_{\theta}(\cdot)\) is determined by the cost function \(L(\cdot,\cdot)\). The second idiom uses the index \(j\) to represent the network's hidden unit total and the number \(n\) to represent the number of units in the hidden layer.