## 3 Technical Design of GSFR-BLS
### Broad Learning Classifier with Enhanced Feature Mapping  
The representation matrix \(R\), constructed from the spatio-temporal dynamic selection of keypoints, serves as the input to the broad learning classifier. \(R\) contains the fused information of the keypoints' spatial and temporal relationships, and it is passed through the feature mapping layer and enhanced feature mapping layer for classification. The feature mapping layer takes the matrix \(R\) as input and transforms it into the feature nodes. By transforming \(R\) into feature nodes, the essential spatial-temporal features across keypoints are captured. This is achieved by performing linear mapping on \(R\) through randomly initialized weights and biases. The process is formulated as  
\[F_{i}=R\cdot W_{Fi}+\beta_{Fi},\quad i=1,\dots,N_{2} \tag{7}\]  
where \(F=[F_{1},\dots,F_{N_{2}}]\) represents the set of feature nodes, each corresponding to a different feature window. Furthermore, \(W_{Fi}\) and \(\beta_{Fi}\) are the random weights and biases for the transformation. This step generates \(N_{2}\) windows, each containing \(N_{1}\) feature nodes. No iterative training is required at this stage, and the transformation is efficient. A combination of sparse feature selection and PCA is proposed, operating on the output from the feature mapping layer \(F\). These methods ensure that only the most relevant features from the matrix \(R\) are retained, leading to improved model efficiency and better classification performance. 1. Sparse feature selection: The L1 regularization to the feature nodes to enforce sparsity and remove noisy or redundant features is applied. The objective function for sparse feature selection is defined as  
\[\min_{W}\|F\cdot W-X\|_{2}^{2}+\lambda\|W\|_{1} \tag{8}\]  
where \(F\) is the feature node matrix obtained from the previous layer, \(W\) are the learned sparse weights, and \(\lambda\) is the regularization parameter controlling the sparsity. 2. Principal Component Analysis: After sparse feature selection, the PCA on the selected features to reduce their dimensionality further is performed. This ensures that the most informative features are preserved while simultaneously lowering computational complexity:  
\[E_{j}=\text{PCA}(F)\cdot W_{Ej}+\beta_{Ej},\quad j=1,\ldots,N_{3} \tag{9}\]  
where \(E=[E_{1},\ldots,E_{N_{3}}]\) represents the enhanced nodes, and \(W_{Ej}\) and \(\beta_{Ej}\) are the random weights and biases for the enhancement mapping. The final output \(Y\) is obtained by concatenating the feature nodes \(F\) and enhanced nodes \(E\) and applying a ridge regression model to learn the weights \(W\) that map these nodes to the target output \(Y\):  
\[Y=[F|E]W \tag{10}\]  
where \([F|E]\) is the concatenation of the feature and enhanced nodes, and \(W\) represents the learned weights used to map the features to the output labels. The learned weights \(W\) are obtained using the Moore-Penrose pseudoinverse:  
\[W=[g(F|E)]^{+}Y \tag{11}\]  
where \(g(x)\) denotes the activation function applied to the feature and enhanced nodes, and \([g(F|E)]^{+}\) is the pseudoinverse of the transformed feature space. During inference, the matrix \(R\) is passed through the same feature mapping layer and enhanced feature mapping layer. The saved weights \(W\) are then used to compute the classification result, providing real-time predictions with high accuracy. The broad learning classifier (BLC) produces an output matrix \(Y\) based on the input representation matrix \(R\). This output \(o(t)\) represents the raw classification probabilities, which will undergo a smoothing process to stabilize the predictions. The final classification result \(o(t)\) at frame \(t\) is computed as  
\[o(t)=\text{softmax}(Y_{t}) \tag{12}\]  
where \(Y_{t}\) is the output from the BLS at frame \(t\), and the softmax function converts the output logits into classification probabilities for the respective classes.