## 2 Proposed Method
### Traffic Sign Classification
#### 2.3.2 Numerical Algorithm  
To find the sparse representation, \(\mathbf{C}\), of each traffic sign candidate in (6), we provide a fast numerical solution based on the alternating direction method of multipliers (ADMM) [16]. To do so, we introduce auxiliary variables in the objective function to simplify the optimization model and derive its augmented Lagrangian. We define a vector \(\mathbf{m}\in\mathbb{R}^{t}\) such that \(\mathbf{m}_{i}=\text{argmax}|\mathbf{C}_{i}(:)|\) and replace the second term of (6a) with \(\lambda\mathbf{1}_{t}^{\top}\mathbf{m}\). To ensure the equivalence of optimization problems, we impose \(\mathbf{m}\otimes\mathbf{1}_{t}\mathbf{1}_{t}^{\top}\geq\mathbf{C}\) as an inequality constraint and provide a non-negative slack matrix \(\mathbf{U}\in\mathbb{R}^{(lt)\times l}\)to compensate the difference between the two sides of this inequality constraint. Therefore, \(\mathbf{m}\otimes\mathbf{1}_{l}\mathbf{1}_{l}^{\top}=\mathbf{C}+\mathbf{U}\). After this simplification, we write the augmented Lagrangian as:  
\[ \begin{split}\mathcal{L}_{\mu}(\mathbf{C},&\mathbf{U},&\hat{\mathbf{C}},\hat{\mathbf{U}},\mathbf{\Lambda}_{1},\mathbf{ \Lambda}_{2})=\left\|\mathbf{X}_{j}-\mathbf{D}\mathbf{C}\right\|_{\mathrm{F}}^ {2}+\frac{\lambda}{l^{2}}\Sigma_{(lt)}^{\top}(\mathbf{C}+\mathbf{U})\mathbf{1 }_{l}\\ &+\frac{\mu_{1}}{2}\left\|\mathbf{C}-\hat{\mathbf{C}}+\frac{ \mathbf{\Lambda}_{1}}{\mu_{1}}\right\|_{\mathrm{F}}^{2}+\frac{\mu_{2}}{2} \left\|\mathbf{U}-\hat{\mathbf{U}}+\frac{\mathbf{\Lambda}_{2}}{\mu_{2}}\right\| _{\mathrm{F}}^{2}\end{split}  \tag{7}\]  
where \(\mathbf{U}\in\mathbb{R}^{(lk)\times l}\) is a non-negative slack matrix, \(\hat{\mathbf{C}}\) and \(\hat{\mathbf{U}}\) are auxiliary variables with the same dimension as \(\mathbf{U}\), \(\mu_{1}\) and \(\mu_{2}\) are positive augmented Lagrangian parameters, and \(\mathbf{\Lambda}_{1},\mathbf{\Lambda}_{2}\in\mathbb{R}^{(lt)\times l}\) are the Lagrangian multipliers. Without loss of generality, we assume \(\mu_{1}=\mu_{2}=\mu\). Given an initialization for \(\hat{\mathbf{C}}\), \(\hat{\mathbf{U}}\), \(\mathbf{\Lambda}_{1}\), and \(\mathbf{\Lambda}_{2}\), the ADMM method is used to solve (7) via multiple iterations:  
\[ \begin{split}(\mathbf{C}^{i+1},\mathbf{U}^{i+1}):=& \operatorname*{arg\,min}_{\mathbf{C},\mathbf{U}\in\mathbb{R}^{(lt) \times l}}\mathcal{L}_{\mu}(\mathbf{C},\mathbf{U},\hat{\mathbf{C}}^{t},\hat{ \mathbf{U}}^{i},\mathbf{\Lambda}_{1}^{i},\mathbf{\Lambda}_{2}^{i})\\ &\text{subject to}\quad\mathbf{E}(\mathbf{C}+\mathbf{U})=\frac{ \mathbf{I}_{l}\otimes\mathbf{1}\mathbf{1}_{l}^{\top}}{l}(\mathbf{C}+\mathbf{U} )\\ (\hat{\mathbf{C}}^{i+1},\hat{\mathbf{U}}^{i+1}):=& \operatorname*{arg\,min}_{\mathbf{C},\mathbf{U}\in\mathbb{R}^{(lt) \times l}}\mathcal{L}_{\mu}(\mathbf{C}^{i+1},\mathbf{U}^{i+1},\hat{\mathbf{C}},\hat{\mathbf{U}},\mathbf{\Lambda}_{1}^{i},\mathbf{\Lambda}_{2}^{i})\\ &\text{subject to}\quad\hat{\mathbf{C}}\geq 0,\\ &\mathbf{1}_{(lt)}^{\top}\hat{\mathbf{C}}=\mathbf{1}_{l}^{\top}, \\ &\hat{\mathbf{U}}\geq 0\end{split}  \tag{9}\]  
\[ \begin{split}\mathbf{\Lambda}_{1}^{i+1}&=\mathbf{ \Lambda}_{1}^{i}+\mu(\mathbf{C}^{i+1}-\hat{\mathbf{C}}^{i+1})\\ \mathbf{\Lambda}_{2}^{i+1}&=\mathbf{\Lambda}_{2}^{i}+ \mu(\mathbf{U}^{i+1}-\hat{\mathbf{U}}^{i+1})\end{split}  \tag{10}\]  
To solve (8), we first stack the \(i\)th rows of \(\mathbf{C}\) and \(\mathbf{U}\) and construct \(\{\mathbf{z}_{i}\}_{i=1}^{lt}\), where \(\mathbf{z}_{i}\in\mathbb{R}^{2l}\). We then divide (8) into \(lt\) equality constrained quadratic problems. Each problem is solved analytically by writing Karush-Kuhn-Tucker (KKT) conditions. Using the solution of (8), we solve the optimization problem in (9) by splitting it into two separate subproblems. The first subproblem is over \(\hat{\mathbf{C}}\) and consists of \(l\) independent Euclidean norm projections onto the probability simplex. The second subproblem is over \(\hat{\mathbf{U}}\) and consists of \(l\) independent Euclidean norm projections onto the non-negative orthant.