## 4 Application to Classification Problems
### Experiments
For \(k=0,1,2,\cdots\), compute**  
\[\mathbf{x}^{(k+1)}= \Big{[}\mathbf{I}_{n}-\frac{1}{\iota}\big{(}\mathbf{A}^{\top}\mathbf{A}- \lambda\mathbf{W}^{\top}\mathbf{B}^{\top}\mathbf{B}\mathbf{W}\big{)}\Big{]}\mathbf{x}^{(k)}-\frac{ \lambda}{\iota}\mathbf{W}^{\top}\mathbf{B}^{\top}\mathbf{B}\mathbf{u}^{(k)}-\frac{\lambda}{ \iota}\mathbf{W}^{\top}\mathbf{v}^{(k)}+\frac{1}{\iota}\mathbf{A}^{\top}\mathbf{y},\] \[\mathbf{u}^{(k+1)}= \text{Prox}_{\frac{\lambda}{\parallel}\cdot\parallel_{2,1}}\Big{[} \frac{2\lambda}{\tau}\mathbf{B}^{\top}\mathbf{B}\mathbf{W}\mathbf{x}^{(k+1)}-\frac{\lambda}{ \tau}\mathbf{B}^{\top}\mathbf{B}\mathbf{W}\mathbf{x}^{(k)}+(\mathbf{I}_{n}-\frac{\lambda}{\tau}\bm {B}^{\top}\mathbf{B})\mathbf{u}^{(k)}\Big{]},\] \[\mathbf{v}^{(k+1)}= 2\mathbf{W}\mathbf{x}^{(k+1)}-\mathbf{W}\mathbf{x}^{(k)}+\mathbf{v}^{(k)}-\text{Prox}_ {\parallel\cdot\parallel_{2,1}}\Big{(}2\mathbf{W}\mathbf{x}^{(k+1)}-\mathbf{W}\mathbf{x}^{(k)}+ \mathbf{v}^{(k)}\Big{)}\]  
until the stopping criterion is fulfilled. **3. Compute the class label \(i^{\star}\) of \(\mathbf{y}\) by**  
\[i^{\star}=\arg\min_{i}\|\mathbf{y}-\mathbf{A}_{i}x_{i}^{(k+1)}\|_{2}. \tag{23}\]  
**Output:** The class label \(i^{\star}\) corresponding to \(\mathbf{y}\). **Algorithm 1**The proposed group-sparsity enhanced classification algorithm  
Figure 2: Estimated sparse coefficients \(\mathbf{\hat{x}}\) by GSC and proposed method respectively. Table 1 summarizes the recognition accuracy of GSC and the proposed method with \(\mathbf{W}=\mathbf{I}_{n}\). The training set includes digits \"0\" through \"4\" \(\beta\) samples per class and \"5\" through \"9\" \(\alpha\) samples per class. Through numerical experiments, we found that GSC with \(\lambda=\lambda_{\text{GSC}}=1.5\) and the proposed method with \(\lambda=\lambda_{\text{prop}}=3\) perform well on this dataset. We also experimented the proposed method of using \(\lambda_{\text{GSC}}\), which did not degrade too much compared with using \(\lambda_{\text{prop}}\). We see that the GSC model degrades when the training set is unbalanced, and the proposed method outperforms GSC especially in such case. Next, we conduct the experiments on a classic face dataset to verify the validity of the proposed linearly involved model by setting the weight matrix \(\mathbf{W}\) according to (9). The ORL Database of Faces [43] contains 400 images from 40 distinct subjects (10 images per subject) with variations in lighting, facial expressions (open or closed eyes, smiling or not smiling) and facial details (glasses or no glasses). In our experiments, following [44], all images were downsampled from \(112\times 92\) to \(16\times 16\) and then formed 256-D vectors. The number of training samples for each class is not necessarily equal, which varies from 4 to 8 (test set is fixed to 2 images per class). We compared the proposed model (20) (\(\mathbf{W}=\mathbf{I}_{n}\) and \(\mathbf{W}\) by (9) respectively) with GSC [10] and WGSC [11]. In order to achieve the overall convexity, we set \(\mathbf{B}=\sqrt{\theta/\lambda}\mathbf{AW}^{-1}\), \(0\leq\theta\leq 1\) and fix \(\theta=0.9\) for proposed method. Settings of \((\iota,\tau,\kappa)\), initial estimate and stopping criterion are the same as those in the previous experiment. When the parameter \(\lambda\) is assigned too small, the obtained coefficient vector is not group sparse; when the parameter \(\sigma_{1}\) or \(\sigma_{2}\) is assigned too small, the information of locality or similarity plays a decisive role. We found that \(\lambda=0.05\) for \(\ell_{2,1}\) regularizer-based methods (i.e., GSC and WGSC), \(\lambda=0.2\) for the proposed method and \(\sigma_{1}\in[2,4],\sigma_{2}\in[0.5,2]\) for weights involved methods (i.e., WGSC and proposed method with \(\mathbf{W}\) by (9)) work well on this dataset. Figure 3 shows a classification result of WGSC and proposed method (\(\mathbf{W}\) by (9)) (both with \(\sigma_{1}=4,\sigma_{2}=2\))) when training set is unbalanced (20 subjects have 8 samples per class and the others have 6 samples per class).