# MCRCNet: A Bearing Fault Diagnosis Method for Unknown Faults Based on Transfer Learning
## 2 Preliminaries  
### Reconstruction Convolution  
The convolution operation of deep learning shows its advantages in feature extraction, but the traditional convolution method has large redundancy, which causes the model to extract too many redundant features, and these redundant features also affect the final accuracy of the model. Therefore, researchers have begun to try to improve the convolution operation in order to achieve the purpose of reducing redundant features in the spatial and channel dimensions. The convolutional reconstruction method based on space and channel reconstruction is a typical representative. This method has two parts, a spatial reconstruction unit and a channel reconstruction unit, and enhances the feature expression ability by operating in two aspects: space and channel, as shown in Figures 1 and 2. The spatial reconstruction unit primarily addresses the issue of spatial feature redundancy. The method it employs involves using group normalization to evaluate the information content of the feature maps and separates the feature maps based on the amount of information. Specifically, given an intermediate feature map \(X\in R\), the input feature \(X\) is first normalized by subtracting the mean \(\mu\) and dividing by the standard deviation \(\sigma\), as shown in the following equation:  
Figure 1: Spatial reconstruction unit. \[X_{out}=\text{GN}(X)=\gamma\frac{X-\mu}{\sqrt{\sigma^{2}+\epsilon}}+\beta. \tag{1}\]  
In the equation, the mean and standard deviation of \(X\) are denoted by \(\mu\) and \(\sigma\). \(\epsilon\) is a small positive constant. The purpose of adding it is to make the division more stable. For affine transformations, we use \(\gamma\) and \(\beta\), which are trainable. For the measurement of pixel variance, we use the parameter \(\gamma\) from the GN layer. A larger value of this parameter indicates richer spatial information. Subsequently, through the sigmoid function, the re-weighted feature map is mapped within the range (0, 1). The gating method uses a threshold: values greater than the threshold are set to 1, corresponding to \(W1\); otherwise, to \(W2\). The specific process is described in the following equation:  
\[W=\text{Gate}(\text{Sigmoid}(W_{\gamma}(\text{GN}(X)))). \tag{2}\]  
Finally, the input feature \(X\) is multiplied by \(W1\) and \(W2\), respectively, to obtain two weighted features: the feature with more information \(X_{1}^{\omega}\) and the feature with less information \(X_{2}^{\omega}\). In this way, the input features are successfully divided into two parts: \(X_{1}^{\omega}\) has informative and expressive spatial content, while \(X_{2}^{\omega}\) has almost no information and is considered redundant. The reconstruction operation process adds the informative features to the less informative ones to generate more informative features, thus saving space. The cross-reconstruction operation is used to fully combine the two different weighted information features to enhance the information flow between them. Then the cross-reconstructed features \(X^{\omega 1}\) and \(X^{\omega 2}\) are concatenated to obtain the spatially fine feature map \(X^{\omega}\). The subsequent calculation process is shown in the equation below  
\[\left\{ \begin{array}{c}X_{1}^{\omega}=W_{1}\otimes X\\ X_{2}^{\omega}=W_{2}\otimes X\\ X_{11}^{\omega}\oplus X_{22}^{\omega}=X^{\omega 1}\\ X_{21}^{\omega}\oplus X_{12}^{\omega}=X^{\omega 2}\\ X^{\omega 1}\cup X^{\omega 2}=X^{\omega 1}\end{array} \right., \tag{3}\]  
where \(\otimes\) is multiplication by element, \(\oplus\) is summation by element, and \(\cup\) is concatenation. After the spatial reconstruction unit is applied to the intermediate input feature \(X\), it not only separates the information features from the features with less information, but also reconstructs them to enhance the representative features and suppress the redundant features in the spatial dimension. However, the spatial fine feature map \(X^{\omega}\) is still redundant in the channel dimension.