算法基于ADMM对(3)式的求解

Initialize ${\mathcal{L}}_0={\mathcal{Y}}_0={\mathcal{W}}_0=0,\rho =1.1,{\alpha }_0={\beta }_0=1\text{e}-3,\epsilon =1\text{e}-8$

While not converged do

1: Update ${\mathcal{E}}_{k+1}$ by

${\mathcal{E}}_{k+1}=\underset{\epsilon }{\arg \min }\lambda {\Vert \mathcal{E}\Vert }_1+\frac{\beta }{2}{\Vert \mathcal{E}-\left({\mathcal{X}}_k-{\mathcal{L}}_k+{\beta }^{-1}{\mathcal{Y}}_k\right)\Vert }_F^2$

2: Update ${\mathcal{L}}_{k+1}$ by

$\begin{array}{l}{\mathcal{L}}_{k+1}=\underset{L}{\arg \min }{\Vert \mathcal{L}\Vert }_{\ast }+\gamma \left({\Vert \mathcal{L}\Vert }_F^2-{\Vert {\mathcal{A}}_l\ast \mathcal{L}\Vert }_F^2\right)+\frac{\alpha }{2}{\Vert \mathcal{L}-{\mathcal{Z}}_k+{\alpha }^{-1}{\mathcal{W}}_k\Vert }_F^2\\ +\frac{\beta }{2}{\Vert \mathcal{L}-{\mathcal{X}}_k+{\mathcal{E}}_{k+1}-{\beta }^{-1}{\mathcal{Y}}_k\Vert }_F^2\end{array}$ ;

3: Update ${\mathcal{Z}}_{k+1}$ by

${\mathcal{Z}}_{k+1}=\text{arg}\underset{\mathcal{Z}}{\min }\langle {\mathcal{W}}_k,{\mathcal{L}}_{k+1}-\mathcal{Z}\rangle +\frac{\alpha }{2}{\Vert {\mathcal{L}}_{k+1}-\mathcal{Z}\Vert }_F^2-Tr\left({\mathcal{A}}_l\ast \mathcal{Z}\ast {\mathcal{B}}_l^{\text{T}}\right)$ ;

4: ${\mathcal{Y}}_{k+1}={\mathcal{Y}}_k+\beta \left({\mathcal{X}}_{k+1}-{\mathcal{L}}_{k+1}-{\mathcal{E}}_{k+1}\right)$ ;

5: ${\mathcal{W}}_{k+1}={\mathcal{W}}_k+\alpha \left({\mathcal{L}}_{k+1}-{\mathcal{Z}}_{k+1}\right)$ ;

6: $k=k+1$ ;

7: Check the convergence conditions

${\Vert {\mathcal{L}}_{k+1}-{\mathcal{L}}_k\Vert }_{\infty }\le \epsilon ,{\Vert {\mathcal{E}}_{k+1}-{\mathcal{E}}_k\Vert }_{\infty }\le \epsilon$ ;

${\Vert {\mathcal{L}}_{k+1}+{\mathcal{E}}_{k+1}-\mathcal{X}\Vert }_{\infty }\le \epsilon$ ;

end while.