## 3 Methodology
### Spatial Transformer
#### 3.3.2 Improved Graph Wavelet Neural Network  
Early works of convolution on graphs tended to utilize graph Fourier transform as a means of projecting graph signals into the spectral domain. The technique relies on the eigenvectors of the Laplacian matrix to facilitate the aggregation of signals on the graph. In the graph Fourier transform, the convolution operation can be described as [27]:  
\[x*_{g}y=U\left(\left(U^{T}y\right)\odot\left(U^{T}x\right)\right) \tag{3}\]  
Figure 2: Architecture of improved spatial transformer. where \(x\) denotes the features on graph, \(*_{g}\) denotes the convolution operator, \(y\) denotes the convolution kernel, \(\odot\) denotes the element-wise Hadamard product, and \(U\) denotes the eigenvectors of the Laplacian matrix. Subsequently, it is feasible to replace \(U^{T}y\) by a diagonal matrix \(g_{\theta}\) and \(\odot\) by matrix multiplication. So we rewrite Equation (3) as \(Ug_{\theta}U^{T}x\). Many kinds of spectral graph convolution neural networks (e.g., ChebNet, GCN) are based on this formulation. STTNs adopts ChebNet to extract the fixed spatial dependencies. [PERSON] et al. pointed out that graph convolution based on Fourier transform suffers from a restriction in graph flexibility that hinders the ability to establish suitable convolution on the graph. Thus, they proposed the GWNN, which aggregates features not only locally but flexibly. Taking the GCN (a simplified version of ChebNet) as an example, the GCN aggregates the graph signals via a Laplacian renormalization trick: \(\tilde{D}^{-\frac{1}{2}}A\tilde{D}^{-\frac{1}{2}}\), where \(\tilde{A}=A+I_{N}\), \(\tilde{D}_{ii}=\sum_{j=0}^{N}\tilde{A}_{ij}\), thus, its graph signal passing process can be described as:  
\[U(1-\Lambda)U^{T} \tag{4}\]  
where \(U\) is the matrix of eigenvectors of the Laplacian matrix \(L=I_{N}-\tilde{D}^{-\frac{1}{2}}A\tilde{D}^{-\frac{1}{2}}\) and \(\Lambda\) is the diagonal matrix of eigenvalues. The signal passing can be described as \(1-\lambda\) in the spectral domain so that it is fixed in the first-order neighborhood, as shown in Figure 2(a) [35]. In the present study, we adopt the GWNN to extract the fixed spatial dependencies and improve it by making the scaling parameter \"\(s\)\" learnable, enabling multi-scale graph signal passing, as shown in Figure 4. Analogous to the graph Fourier transform, GWNN projects graph signals into the spectral domain by a set of graph wavelet basis. The graph wavelet basis are defined as \(\psi_{s}=\left(\psi_{\text{s1}},\psi_{\text{s2}},\dots,\psi_{\text{sn}}\right)\), where \(\psi_{si}\) is equivalent to a signal  
Figure 3: The graph signals of different frequencies passing in the spectral domain during feature aggregation.The right part is the graph signals passing of center node 5 in the spatial domain and the colorbar denotes the weights of nodes during feature aggregation. The subfigure **(a,b)** shows the graph signals aggregation of GCN and GWNN respectively. The \(s\) in subfigure **(b)** is the scaling parameter. on a graph diffused away from the node \(i\), and \(s\) is the scaling parameter of the graph wavelets [35]. \(\psi_{si}\) can be obtained using the following formulation:  
\[\psi_{s}=UG_{s}U^{T} \tag{5}\]  
where \(G_{s}=diag(g(s\lambda_{1},s\lambda_{2},\dots,s\lambda_{n}))\) is the scaling matrix and \(g(s\lambda_{i})=e^{s\lambda_{i}}\), \(\lambda_{i}\) is the eigenvalue of the Laplacian matrix. Meanwhile, \(\psi_{s}^{-1}\) can be obtained by replacing \(g(s\lambda_{i})\) with \(g(-s\lambda_{i})\) corresponding to a heat kernel [36]. Otherwise, \(\psi_{s}\) and \(\psi_{s}^{-1}\) can be efficiently calculated using a Chebyshev polynomials approximation [35; 37]. By adjusting the scaling parameters, the neighborhood of feature aggregation can expand or shrink flexibly, as shown in Figure 3b.