# Adaptive Channel-Enhanced Graph Convolution for Skeleton-Based Human Action Recognition
## 3 Method
### Preliminaries  
**Notation. ** Human joints are used as graph nodes, and the human skeleton is used as an edge connecting the nodes in the skeleton data of human actions to create an undirected graph. The skeleton diagram of human action is represented as \(G=(V,E,X)\), where \(V=\{v_{1},v_{2},\ldots,v_{N}\}\) is a set of \(N\) joint points. \(E\) is the edge set of the skeleton graph. \(X\) is the feature set of \(N\) joint points, where \(X\)\(\in\)\(R^{N\times C}\). \(C\) is the number of characteristic channels. The adjacency matrix \(A\)\(\in\)\(R^{N\times N}\) represents the predefined connection relationships of human joints. The elements \(a_{i,j}\) and \(a_{j,i}\) in the adjacency matrix represent the connection relationships between joint points \(v_{i}\) and \(v_{j}\). When joint points \(v_{i}\) and \(v_{j}\) are connected, \(a_{i,j}\), \(a_{j,i}\
eq 0\). A skeleton sequence \(Z=\{z_{1},z_{2},\ldots,z_{T}\}\) in inputted, where \(z_{t}\) is the 3D coordinates of every joint point in the skeleton graph at t and indicates the spatial state of human action in the \(t\) time frame. **Spatial Temporal Graph Convolutional Network. ** The skeletal data for human action recognition comprises the spatial and temporal attributes of the action. The ST-GCN model efficiently captures the spatio-temporal features of the action by employing a hierarchical network that alternates between a GCN and TCN. Spatial convolution can be divided into two steps: feature embedding and feature aggregation. Feature embedding converts the input feature \(X\) into a high-dimensional feature \(\widetilde{X}\) with the formula shown in Equation (1):  
\[\widetilde{X}=conv2D_{1\times 1}(X) \tag{1}\]  
In spatial graph convolution, the node features can be aggregated with those of its neighboring nodes by the product of the adjacency matrix \(A\) and the node feature matrix \(\widetilde{X}\) after feature transformation. In order to prevent the influence of the out-degree and in-degree of the nodes, it is usually necessary to perform symmetric normalization on the adjacency matrix first. The symmetric normalization formula is shown in Equation (2):  
\[\widetilde{A}=\Lambda^{-\frac{1}{2}}(A+I)\Lambda^{-\frac{1}{2}} \tag{2}\]  
The matrix \(\Lambda\) represents the degrees of the nodes, which is defined as \(\Lambda=\text{diag}(d_{0},\ldots,d_{N-1})\). \(I\) is the self-connecting node matrix. After the normalization of the adjacency matrix \(A\), the aggregation of the nodes is shown in Equation (3):  
\[\hat{X}=\widetilde{A}\widetilde{X} \tag{3}\]  
In each layer of spatial graph convolution, the new feature representation \(\hat{x}_{i}^{(l+1)}\) of node \(v_{i}\) is obtained by aggregating the current feature \(\hat{x}_{i}^{(l)}\) of node \(v_{i}\) with the \(\hat{x}_{j}^{(l)}\) of its neighbor node \(v_{j}\), as shown in Equation (4):  
\[\hat{x}_{i}^{(l+1)}=\sigma(\sum_{j\in N(i)}\widetilde{A}_{ij}^{(l)}\hat{x}_{j }^{(l)}w^{(l)}) \tag{4}\]  
\(N\)_(i)_ denotes the set of neighbor nodes of node \(v_{i}\), containing node \(v_{i}\) itself, and \(w\) denotes the weight. Through the aggregation and updating of nodes, the spatial graph convolution of each layer gradually extracts richer graph structure information. As the number of network layers increases, the node features gradually integrate the influence from more distant neighbors, which makes the feature representation extend from the local neighborhood to a larger range of graph structures. The feature extraction process of spatial graph convolution can be represented by Equation (5):  
\[\hat{X}^{(l+1)}=\varphi(BN(\widetilde{A}^{(l)}\hat{X}^{(l)}W^{(l)})) \tag{5}\]  
where \(\varphi\) is the activation function. \(BN(\cdot)\) is the normalized processing function, and \(W\) is the linear mapping matrix. In order to obtain the structural information of the temporal dimension in the human skeleton action samples, the temporal convolution network (TCN) employs a one-dimensional convolution kernel to convolve the sequence data along the temporal dimension.