# EEG Emotion Classification Based on Graph Convolutional Network
## 3 Method
### Graph Learning Block  
Many works suggest that the correlation of regional brain activation plays a critical role in the processing of emotions [12; 39]. Thus, with given features of an individual region, we seek a method that allows the study of how these regions relate to one another. In neuroscience research, graph theoretic analyses are frequently employed to investigate functional brain networks from a topological standpoint since they provide a systematic approach to analyze brain structure by mapping neural elements into nodes and their connections into edges [40; 41; 42]. Following such an idea, we feed the regional features derived from the region encoder into a graph convolutional network (GCN). We employ the spectral approach to define the convolutional filter instead of spatial approach since it is the more dominant method for dealing with signal processing [43].  
The regional features derived from Section 3.2 are used as nodes to formulate a graph \(\mathcal{G}\). Then, assume we have the graph \(\mathcal{G}\) with adjacency matrix \(A\in\mathbb{R}^{n\times n}\) and degree matrix \(D\in\mathbb{R}^{n\times n}\). We can derive the Laplacian matrix by \(L=D-A\), or the normalized version \(L=I-D^{\frac{1}{2}}AD^{-\frac{1}{2}}\). Since \(L\) will be a real symmetric positive semidefinite matrix, it can be decomposed into  
\[L=U\Lambda U^{T} \tag{6}\]  
via singular value decomposition where \(\lambda=diag(\lambda_{1},\cdots,\lambda_{n})\) is the diagonal matrix of the eigenvalues of \(L\) and \(U\) is the Fourier basis. For a given graph data point \(X\), we can derive its graph Fourier transform and inverse graph Fourier transform:  
\[\hat{X} = U^{T}X, \tag{7}\] \[X = U\hat{X}. \tag{8}\]  
Then, the signal filtered by function \(g_{\theta}(\cdot)\) can be expressed by  
\[Y = g_{\theta}(L)X \tag{9}\] \[= g_{\theta}(U\Lambda U^{T})X\] \[= Ug_{\theta}(\Lambda)U^{T}X,\]  
where \(g_{\theta}(\Lambda)\) can be expressed as:  
\[g_{\theta}(\Lambda)=\begin{bmatrix}g(\lambda_{0})&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\ldots&g(\lambda_{N-1})\end{bmatrix}. \tag{10}\]  
Due to difficulties on the large computation caused directly by learning \(g_{\theta}(\Lambda)\), we additionally adopt Chebyshev expansion [44], which is formulated as following:  
\[T_{0} = 1,T_{1}=x, \tag{11}\] \[T_{k}(x) = 2 xT_{k-1}(x)-T_{k-2}(x), \tag{12}\]  
where \(T_{k}(x)\) is the Chebyshev polynomial with order \(k\). Say we have \(\lambda_{max}\) as the largest eigenvalue of \(L\) and \(I_{n}\in\mathbb{R}^{n\times n}\) is the identity matrix  
where the parameter \(\theta\in\mathbb{R}^{K}\) presents the Chebyshev coefficients. Similarly, we can derive \(g_{\theta}(L)=\sum_{k=0}^{K-1}\theta_{k}T_{k}(L)\) with \(\bar{L}=\frac{2L}{\lambda_{max}}-I_{n}\). Back to the filtering operation, we now have:  
\[Y=g_{\theta}(L)X=\sum_{k=0}^{K-1}\theta_{k}T_{k}(\bar{L})X. \tag{14}\]  
Additionally, to allow the network to learn the connectivity among brain regions in a flexible way, we utilize a dynamical graph convolutional network where the adjacency matrix can be learned.  
The detail of the structure of the graph learning block is provided in Figure 3. Finally, the output embedding will generate the final results for predicting the label.