## 2 Notation and Background
### The LCLE-DL Algorithm  
To improve the classification performance, the LCLE-DL algorithm [7] takes both the locality and label information of dictionary atoms into account in the learning process. This algorithm firstly constructs the locality constraint to ensure that similar profiles have similar atoms, then establishes the label embedding constraint to encourage the atoms of the same class to have similar profiles. The objective function of LCLE-DL is defined as follows.  
\[\begin{split}\min_{D_{p},Z_{p},V_{p},L}&\|X-D_{p}Z_{ p}\|_{2}^{2}+\alpha Tr\big{(}Z_{p}^{\mathrm{T}}LZ_{p}\big{)}+\|X-D_{p}V_{p}\|_{2}^{2}+ \beta Tr\big{(}V_{p}^{\mathrm{T}}UV_{p}\big{)}\\ &+\gamma\|Z_{p}-V_{p}\|_{2}^{2},\ \ \ s.t.\|d_{j}\|^{2}=1,\ j=1, \cdots,K\end{split} \tag{1}\]  
where \(Z_{p}\in\mathbf{R}^{K_{p}\times N}\) and \(V_{p}\in\mathbf{R}^{K_{p}\times N}\) denote the locality-based and the label-based coding coefficients, \(L\in\mathbf{R}^{K_{p}\times K_{p}}\) is the graph Laplacian matrix that is constructed by the atom's similarity in the dictionary \(D_{p}\in\mathbf{R}^{m\times K_{p}}\), \(U\in\mathbf{R}^{K_{p}\times K_{p}}\) is the scaled label matrix which is constructed using the label matrix of the dictionary \(D_{p}\). \(\|X-D_{p}Z_{p}\|_{2}^{2}\) combined with the second term encodes the reconstruction under the locality constraint; \(\|X-D_{p}V_{p}\|_{2}^{2}\) combined with fourth term encodes the reconstruction under the label embedding; \(\|Z_{p}-V_{p}\|_{2}^{2}\) is used to transfer the label constraint to the locality constraint. \(\alpha,\ \beta\) and \(\gamma\) are the regularization parameters; the constraint on the atoms can avoid the scaling issue.  
The LCLE-DL algorithm first exploits the K-SVD algorithm to learn sub-dictionaries \(D_{i}(i=1,\cdots,C)\) using with the training samples \(X_{i}\). Similar to the label matrix of the training samples \(Y\), the label matrix of the dictionary \(D_{p}\) can be obtained as \(B=[b_{1},\cdots,b_{K_{p}}]\in\mathbf{R}^{K_{p}\times C}\). Then a weighted label matrix is constructed by \(G=B(B^{\mathrm{T}}B)^{-\frac{1}{2}}\in\mathbf{R}^{K_{p}\times C}\). Next, the label embedding of atoms is defined as  
\[\min_{V_{p}}Tr(V_{p}^{\mathrm{T}}GG^{\mathrm{T}}V_{p})=\min_{V_{p}}Tr(V_{p}^{ \mathrm{T}}UV_{p}), \tag{2}\]  
where \(U=GG^{\mathrm{T}}\in\mathbf{R}^{K_{p}\times K_{p}}\) is the scaled label matrix of the dictionary \(D_{p}\); the above terms make the coding coefficients \(V_{p}\) have a block-diagonal structure with strong discriminative information.  
The learned dictionary inherits the manifold structure of the data via using the derived-graph Laplacian matrix \(L\), and the optimal representation of the samples can be obtained with the label embedding of dictionary atoms. By combining the double reconstructions, LCLE-DL ensures the label-based and the locality-based coding coefficients are as approximate to each other as possible. However, the locality constraint is imposed on the class-specific dictionary in the LCLE-DL algorithm, which may cause the dictionary atoms from the different classes to be similar, thus the discriminative ability of the dictionary is weakened.