## 3 Double Low-Rank and Sparse Decomposition Model
### Optimization  
As \(\text{rank}(\cdot)\) and \(||\cdot||_{0}\) are not convex, Equation (2) is NP-hard problem. A common heuristic criterion is to replace \(\text{rank}(\cdot)\) and \(||\cdot||_{0}\) are replaced by nuclear norm \(||\cdot||_{*}\) and \(l_{1}\) norm \(||\cdot||_{1}\) respectively. It has been shown that nuclear norm-based models can obtain the optimal low-rank solution in many kinds of applications [27; 28]. Then, Equation (2) can be converted to the following convex surrogate optimization problem:  
\[ \begin{array}{l}\underset{L,S}{\text{min}}\big{(}||L||_{*}+\alpha||S||_{*}+ \beta\text{tr}\big{(}SMS^{T}\big{)}+\gamma||L||_{1}\big{)}\\ \text{s.t.}D=S+L\end{array}  \tag{6}\]  
where \(||\cdot||_{*}\) equals the sum of singular values of matrix; \(||\cdot||_{1}\) equals the sum of the absolute values of each element of matrix. For a matrix \(A=a_{ij}\in\mathbb{R}^{m\times n}\), \(||A||_{p}=\left(\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}|a_{ij}|^{p}\right) ^{\frac{1}{p}}\). To solve Equation (6) efficiently, the alternating direction method (ADM) algorithm [29] can be adopted. By introducing the auxiliary variables \(H\) and \(J\), the augmented Lagrange function is given as follows:  
\[ \begin{array}{l}O(L,S,H,J,Y_{1},Y_{2},Y_{3},\mu)\\ =||L||_{*}+\alpha||S||_{*}+\beta\text{tr}\big{(}HMH^{T}\big{)}+\gamma||J||_{1} \\ \quad+\text{tr}\big{(}Y_{1}^{T}(D-L-S)\big{)}+\frac{\mu}{2}||D-L-S||_{F}^{2} \\ \quad+\text{tr}\big{(}Y_{2}^{T}(H-S)\big{)}+\frac{\mu}{2}||H-S||_{F}^{2}\\ \quad+\text{tr}\big{(}Y_{3}^{T}(J-L)\big{)}+\frac{\mu}{2}||J-L||_{F}^{2}\end{array}  \tag{7}\]  
where \(||\cdot||_{F}^{2}\) denotes the Frobenius norm of matrix, which is defined as the sum of squares of each element of matrix; \(Y_{1}\), \(Y_{2}\) and \(Y_{3}\) are Lagrange multipliers; \(\mu>0\) is a penalty parameter. Therefore, Equation (7) can be converted to the following equivalent optimization problem:  
\[ \begin{array}{l}O(L,S,H,J,Y_{1},Y_{2},Y_{3},\mu)\\ =\frac{1}{2}\Big{|}\bigg{|}D-L-S+\frac{Y_{1}}{\mu}\Big{|}\Big{|}_{F}^{2}+\frac {1}{2}\bigg{|}\bigg{|}H-S+\frac{Y_{2}}{\mu}\Big{|}\Big{|}_{F}^{2}+\frac{1}{2} \bigg{|}\bigg{|}J-L+\frac{Y_{3}}{\mu}\Big{|}\Big{|}_{F}^{2}\\ \quad+\frac{1}{\mu}||L||_{*}+\frac{\alpha}{\mu}||S||_{*}+\frac{\beta}{\mu} \text{tr}\big{(}HMH^{T}\big{)}+\frac{1}{\mu}||J||_{1}\end{array}  \tag{8}\]  
The above optimization problem can be solved by alternately updating one variable while others fixed. The detailed ADM algorithm for proposed DLRSD model is summarized in Algorithm 1. (1) Updating \(H\)  
In order to solve \(H\), the optimal solution can be obtained by Equation (9):  
\[\underset{H}{\text{min}}\Bigg{(}\frac{1}{2}\bigg{|}\bigg{|}H-S+\frac{Y_{2}}{ \mu}\bigg{|}\bigg{|}_{F}^{2}+\frac{\beta}{\mu}\text{tr}\Big{(}HMH^{T}\Big{)} \Bigg{)} \tag{9}\]  
Differentiating it with respect to \(H\), and let it to be zero, therefore  
\[H-S+\frac{Y_{2}}{\mu}+\frac{2\beta}{\mu}HM=0 \tag{10}\]  
The close-form solution can be obtained as follows:  
\[H^{*}=\bigg{(}S-\frac{Y_{2}}{\mu}\bigg{)}\bigg{(}I+\frac{2\beta}{\mu}M\bigg{)} ^{-1} \tag{11}\](2) Updating \(J\)  
In order to solve \(J\), the optimal solution can be obtained by Equation (12):  
\[\min_{J}\Biggl{(}\frac{1}{2}\biggl{|}\biggl{|}L-\frac{Y_{3}}{\mu}-J\biggr{|} \biggl{|}_{F}^{2}+\frac{\gamma}{\mu}||J||_{1}\Biggr{)} \tag{12}\]  
The solution is  
\[J^{*}=\Psi_{\frac{2}{\mu}}\biggl{(}L-\frac{Y_{3}}{\mu}\biggr{)} \tag{13}\]  
where \(\Psi_{\frac{2}{\mu}}(\cdot)\) denotes soft-thresholding shrinkage operator, which is defined as  
\[\Psi_{\frac{2}{\mu}}(T)=\text{sgn}(T)\text{max}\biggl{(}|T|-\frac{\gamma}{\mu},0\biggr{)}=\left\{ \begin{array}{ll}T_{ij}-\frac{\gamma}{\mu}&T_{ij}>\frac{ \gamma}{\mu}\\ 0&-\frac{\gamma}{\mu}\leq T_{ij}\leq\frac{\gamma}{\mu}\\ T_{ij}+\frac{\gamma}{\mu}&T_{ij}<-\frac{\gamma}{\mu}\end{array} \right. \tag{14}\]  
where \(T\) denotes a matrix, \(T_{ij}\) denotes the \((i,j)\)-th element of \(T\), \(\text{sgn}(T)\) is the matrix whose entries are the signs of those of \(T\).