## 2 Proposed Method
### Model
#### 2.1.1 Extract Features Using Auto-Encoder  
Like most unsupervised deep learning models, the auto-encoder seeks to achieve abstract learning of input samples by equating the network's predicted output with the input samples. [PERSON] et al. [2018] suggested the auto-encoder concept first, and then [PERSON] et al. [2018] discussed it. An encoder and a decoder are generally found in an auto-encoder. The encoder translates high-dimensional input samples to low-dimensional abstract representations to minimize sample dimensionality. The decoder translates the  
Figure 1: The basic flow of an unsupervised score-guided network. In this model, the representation learner is simultaneously trained with the scoring network. The scoring network guides the representation learner to learn better discriminative information from the input data. Figure 2: The unsupervised structural damage detection model’s overall schematic diagram. abstract representations into predicted outputs to replicate the input samples. It adjusts the network parameters to minimize the reconstruction error to acquire the best abstract representation of the input samples. Figure 3 depicts the auto-encoder structure. Suppose given an input sample \(X=R^{d\times n}\), where d represents the dimension of the input sample and \(n\) is the number of input samples. The weight matrix and bias of the encoder and decoder are \(w_{m}\), \(w_{d}\) and \(b_{m}\), \(b_{d}\), respectively. \(g(\cdot)\) is the activation function of each layer. The auto-encoder sends samples to the encoder and uses linear mapping and a nonlinear activation function to finish the encoding. \[H=g(W_{m}X+b_{m}) \tag{1}\]  
The decoder then completes the decoding of the encoded features. It obtains the input sample reconstruction \(\hat{X}\). Given the encoding \(H\), \(\hat{X}\) can be thought of as a prediction of \(X\) with the same dimension as \(X\). The decoding procedure is similar to that of encoding. \[\hat{X}=g(W_{d}H+b_{d}) \tag{2}\]  
The \(L_{2}\) loss is commonly employed as a cost function for reconstruction to optimize the weight matrix and biases parameters. It can be stated as follows:  
\[L_{RE}(X,\hat{X})=\frac{1}{n}\sum_{i=1}^{n}\|X_{i}-\hat{X}_{t}\|_{2} \tag{3}\]  
To learn feature representations in the sample data, auto-encoders commonly use a gradient descent technique to backpropagate the error to alter the network parameters and gradually reduce the reconstructed error function by iterative fine-tuning. Assuming that the learning rate is \(\eta\), the auto-connection encoder's weights and bias update formulas are as follows:  
\[W=W-\eta\frac{\partial L_{RE}(X,\hat{X})}{\partial W} \tag{4}\]  
\[b=b-\eta\frac{\partial L_{RE}(X,\hat{X})}{\partial b} \tag{5}\]  
Figure 4 shows how this paper extracts characteristics from response data using a deep auto-encoder with five hidden layers. Deep auto-encoders with multiple hidden layers can learn more complex underlying feature representations of input data mathematically than auto-encoders with only one hidden layer. The encoder is represented by layers \(L_{1}\) to \(L_{4}\), while layers \(L_{4}\) to \(L_{7}\) represent the decoder in Figure 4. The previous layer's output is used as the next layer's input. The encoder weight matrices \(w_{1}\), \(w_{2}\) and \(w_{3}\) are used  
Figure 3: Auto-encoder structure. to learn the final compressed feature representation on the \(L_{4}\) layer; \(b_{1}\), \(b_{2}\), and \(b_{3}\) are the encoder biases. The decoding matrices \(w_{5}\), \(w_{6}\), \(w_{7}\) and the biases \(b_{5}\), \(b_{6}\), \(b_{7}\) of the decoder are responsible for reconstructing input data on output layer \(L_{7}\) using the learned feature representation on the \(L_{4}\) layer. The function \(g(\cdot)\) aims to strengthen the learning ability of the auto-encoder. In this paper, the autoencoder adopts Equation (3) as the reconstruction loss function.