## 2 Basic Theory  
### Basic Theory of SAE  
Equipment condition monitoring data often has noise signals in the data due to the working environment or operating conditions, which leads to \"impure\" data. To realize the accurate analysis of the data, we need to \"clean\" the data, extract the effective features, and restore the pure data. In practical application, principal component analysis (PCA), as a classical method to improve the signal-to-noise ratio (SNR) and linear dimension reduction, is limited in subspace feature extraction [28], and the determination of principal component weight is often subjective, so it is not suitable for the understanding of nonlinear features. However, [PERSON] [29] in 2006 proposed a low dimensional expression method of learning high-dimensional features by training deep \"autoencoder\" network, which is essentially a signal compression model based on neural network. Autoencoder(AE) compression embodies its advantages in feature compression and provides a new direction for the research in this field. The simplified network structure of shallow AE is shown in Figure 1.  
The AE is a neural network that reconstructs the input signal from the target expression. The purpose is to obtain a dimension reduction feature expression \(H=\{h_{(1)},h_{(2)},h_{(3)},\dots\}\) of the data through training based on the input unlabeled data \(X=\{x_{(1)},x_{(2)},x_{(3)},\dots\}\). The AE encodes the input vector \(X\) to the compression feature of the hidden layer through activation function mapping to express \(H\):  
\[H=f_{(\theta)}(x)=s(Wx+b) \tag{1}\]  
The feature expression \(H\) of the hidden layer is reconstructed by mapping and decoding as follows:  
\[\hat{x}=g_{\theta^{\prime}}(H)=s(W^{\prime}H+b^{\prime}) \tag{2}\]  
where, \(\theta=(W,b)\) and \(\theta^{\prime}=(W^{\prime},b^{\prime})\) are coding model parameters and decoding model weight parameters respectively; \(W\) and \(W^{\prime}\) are encoding weight matrix and decoding  
Figure 1: Simplified network structure of autoencoder (AE). weight matrix respectively; \(b\) and \(b^{\prime}\) are bias vectors; \(s\) function is the activation function, and the general expression is \(s(u)=sigmoid(u)=1/(1+exp(-u))\). The reconstruction result \(\hat{x}\) cannot reproduce the input \(x\) completely and accurately. Our goal is to find the minimum reconstruction error of parameters \(\theta\) and \(\theta^{\prime}\), and to use loss function to represent the training effect and to minimize the loss function. At this time, the common characteristics of the input information \(x\) and the reconstruction information \(\hat{x}\) are extracted to the maximum. Generally, there are square error loss function and cross entropy loss function, which are respectively expressed as:  
\[L(x,\hat{x})=\frac{1}{n}\left\|x-\hat{x}\right\|^{2} \tag{3}\]  
\[L(x,\hat{x})=-\sum_{i=1}^{n}[x_{i}lg(\hat{x})+(1-x_{i})lg(1-\hat{x})] \tag{4}\]  
The optimization function is expressed as:  
\[(\theta^{*},\theta^{*^{\prime}})=argmin\frac{1}{n}\sum_{i=1}^{n}L(x_{i},\hat{ x}_{i})=argmin\frac{1}{n}\sum_{i=1}^{n}L(x_{i},g_{\theta^{\prime}}[f_{ \theta}(x_{i})]) \tag{5}\]  
Considering that there will be noise information in the actual operation process of bearings, this paper uses the de-noising autoencoder (DAE) to add noise to the original features, that is, the original input vector \(\mathbf{x}\) is added with noise to get \(\widetilde{\mathbf{x}}\), the unit is randomly selected according to a certain proportion and forced to be set to 0, and then the de-noised data \(\widetilde{x}\) is trained for encoding and decoding.