# Multi-Perception Graph Convolution Transfer Network Bearing Fault Diagnosis Method
## 2 Preliminaries
### Graph Convolution Network  
Convolution neural networks (CNNs) have been extensively employed in various areas, including image recognition and natural language processing [18]. However, they are limited to processing normal Euclidean data, such as grids and sequences, and are unable to process non-Euclidean data, like knowledge graph data, efficiently. Traditional CNNs have trouble processing such network structure data since the graph's structure is typically quite unpredictable, each node's surrounding structure can be distinct, and its associated edges and neighboring nodes differ. The graph convolution neural network (GCN) is a CNN application for non-Euclidean graph-structured data [19]. GCN methods can be divided into two main categories: spectrum based and space based. From the standpoint of graph signal processing, the spectrum-based methods describe graph convolution by introducing a filter. The neighborhood-based feature information is aggregated in the space-based methods' representations of graph convolution. In graph theory, the relationship between nodes can be expressed as a relation graph \(G=G(V,E)\), where \(G\) represents a graph, \(V\) is the set of vertices in the graph \(G\), and \(E\) is the set of edges in the graph \(G\). In a spectrum-based GCN, graphs are assumed to be undirected, and one robust mathematical representation of undirected graphs is the regularized graph Laplacian matrix, as shown below. \[L=D^{-1/2}LD^{-1/2}=I_{N}-D^{-1/2}AD^{-1/2} \tag{1}\]  
where \(I_{N}\) is the identity matrix, \(A\) is the adjacency matrix of the graph, \(D\) is the diagonal matrix, and \(D_{ii}=\sum_{j}(A_{i,j})\). The eigenvalue decomposition of this Laplacian matrix \(L\) is  
\[L=U \begin{pmatrix}\lambda_{1}&&\\ &\ddots&\\ &&\lambda_{n}\end{pmatrix} U^{-1}=U\Lambda U^{-1} \tag{2}\]  
where \(\Lambda=\text{diag}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\) is a diagonal matrix composed of eigenvalues of \(L\), and the orthogonal matrix \(U\) is composed of eigenvectors of \(L\). The basic principle of the GCN is as follows. Taking into account the convolution operation \(g*x\), where \(g\) is the convolution kernel and \(x\) are the eigenvectors of the node,with the Fourier transform, the frequency domain form of the above convolution operation can be expressed as follows:  
\[g*x=U\Big{(}\Big{(}U^{T}g\Big{)}\odot\Big{(}U^{T}x\Big{)}\Big{)} \tag{3}\]  
where \(U^{T}x\) represents the Fourier transform of \(x\), and \(\odot\) is the Hadamard product. \(U^{T}g\) can also be expressed as a diagonal matrix \(U^{T}g=g_{\theta}(\Lambda)\). Formula (3) can be simplified as follows:  
\[g*x=Ug_{\theta}(\Lambda)U^{T}x=Ug_{f}U^{T}x \tag{4}\]  
where \(g_{f}\) represents the map filter of \(N\) parameters to be trained. However, directly training a set of \(N\) parameters is a lot of calculations. To simplify the calculation, [PERSON] et al. (2018) used Chebyshev polynomials to approximate graph convolution filters. In the network, the vertex embedding can be calculated according to the \(K\)-th Chebyshev polynomial, which is represented as follows:  
\[f_{\theta^{\prime}}=\sum_{k=0}^{K-1}\theta_{k}\Lambda^{k}=\sum_{k=0}^{K-1} \theta^{\prime}_{k}T_{k}(\tilde{\Lambda}) \tag{5}\]  
where \(K\) is the largest order of the Chebyshev polynomial, \(\tilde{\Lambda}=\frac{2\Lambda}{\lambda_{\text{max}}}-I_{N}\) is the rescaled eigenvalue, \(\lambda_{\text{max}}\) represents the largest eigenvalue, \(\theta\) is the vector of the coefficients of the polynomial, and \(\theta^{\prime}_{k}\in R^{K}\) is the vector of the Chebyshev coefficients. \(T_{k}(\tilde{\Lambda})\) is a Chebyshev polynomial of order k.  
Following the Chebyshev polynomial approximation of the filter, the graph convolution of node characteristics with the filter may be expressed mathematically as follows:  
\[h=U \begin{bmatrix}\theta^{\prime}_{1}T_{1}\Big{(}\tilde{\Lambda}\Big{)}& \cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\theta^{\prime}_{k}T_{k}\Big{(}\tilde{\Lambda}\Big{)}\end{bmatrix}