# AFGN: Adaptive Filtering Graph Neural Network for Few-Shot Learning
## 2 Materials and Methods
### The Proposed Method
#### 2.2.4 Graph Convolution of AFGN  
Graph Convolutional Neural Network is a type of neural network model specifically designed to handle graph-structured data. The core concept of GCN is to utilize convolution operations to propagate and aggregate information from neighboring nodes, thereby obtaining new representations for each node. Therefore, the multi-layer graph convolution in GCN is generally represented as:  
\[F^{(l+1)} =\sigma(D^{-\frac{1}{2}}A^{(l)}D^{-\frac{1}{2}}F^{(l)}W^{(l)}) \tag{7}\] \[=\sigma[(\mathbb{I}-L^{(l)})F^{(l)}W^{(l)}],\]  
where \(F^{(l+1)}\) denotes the node feature matrix of the \(l\)-th layer with \(F^{0}=F\), \(A^{(l)}\) represents the adjacency matrix of the \(l\)-th layer with \(A^{0}=A\), \(L^{(l)}\) denotes the Laplacian operator of the \(l\)-th layer with \(L^{0}=L\), \(W^{(l)}\) signifies the weight matrix of the \(l\)-th layer, and \(\sigma\) represents the nonlinear activation function. In fact, GCN can be regarded as an implementation or extension of Spectral CNN, the essence of graph convolution is the filtering process of graph signals in the frequency domain. For the node feature matrix \(F\in\mathbb{R}^{n\times l}\) on the graph, where each row represents the feature of a node, each column can be viewed as a graph signal \(x\in\mathbb{R}^{n}\). Based on research on graph signal processing in the frequency domain, we can utilize the graph Laplacian to analyze and process graph signals. Specifically, the Laplace matrix \(L\in\mathbb{R}^{n\times n}\) after eigenvalue decomposition has \(L=U\Lambda U^{T}\), where \(U=[u_{1},u_{2},...u_{n}]\) is composed of the eigenvectors of \(L\), also known as the graph Fourier basis. \(\Lambda=diag[\lambda_{1},\lambda_{1},...\lambda_{n}]\) is a diagonal matrix with diagonal elements being the eigenvalues of \(L\), where \(\lambda_{i}\in[0,2]\). In particular, these eigenvalues \(\{\lambda_{i}\}_{i=1}^{n}\) are also considered as frequencies of the graph. We utilize the graph Fourier transform to convert the image signal \(x\) from the spatial domain to the frequency domain:  
\[\tilde{x}=U^{T}x= \begin{bmatrix}u_{1}^{T}x\\ u_{2}^{T}x\\ \vdots\\ u_{n}^{T}x\end{bmatrix} = \begin{bmatrix}\tilde{x}_{1}\\ \tilde{x}_{2}\\ \vdots\\ \tilde{x}_{n}\end{bmatrix}  \tag{8}\]  
where \(\tilde{x}\) represents the projection of \(x\) in the graph Fourier basis direction \(u_{i}\). Then, we use the inverse graph Fourier transform to convert the result back to the spatial domain \(x=U\tilde{x}\). Therefore, utilizing the basis transformation of the graph Fourier, the filtering of graph signals \(x\) in frequency domain is defined as:  
\[x*_{G}g=Ug_{(\theta)}U^{T}x, \tag{9}\]  
\(g_{(\theta)}\) is a trainable parameter with respect to \(\Lambda\). To enhance the spatial locality of the convolution, we approximate it with \(K\)-th order polynomials: where \(*_{G}\) represents the graph convolution operator, \(g_{(\theta)}\) denotes the convolution kernel in the frequency domain, also viewed as a joint filter. \(g_{(\theta)}\) is a trainable parameter with respect to \(\Lambda\). To enhance the spatial locality of the convolution, we approximate it with \(K\)-th order polynomials:  
\[g_{\theta}=\sum_{k=0}^{K-1}\alpha_{k}[\mathcal{P}(\Lambda)]^{k}, \tag{10}\]  
where \(K<n\), \(\alpha_{k}\) is the polynomial coefficient and \(\mathcal{P}(\Lambda)=diag[p(\lambda_{1}),p(\lambda_{2}),...p(\lambda_{n})]\) is a function related to \(\lambda_{i}\). When \(K=2\), we obtain the expansion of Formula (9):  
\[ \begin{split} x*_{G}g&=Ug_{\theta}U^{T}x\\ &=U[\alpha_{0}+\alpha_{1}p(\Lambda)]U^{T}x\\ &=\{\alpha_{0}\mathbb{I}+\alpha_{1}[p(\lambda_{1})u_{1}u_{1}^{T}+ p(\lambda_{2})u_{2}u_{2}^{T}+...p(\lambda_{n})u_{n}u_{n}^{T}]\}x,\end{split}  \tag{11}\]  
when \(\lambda_{1}\) is low-frequency, the corresponding frequency component of \(\lambda_{1}\) is the low-frequency component, which represents the part where the graph signal changes smoothly.