Algorithm 1. Two-step approach for (13).

Input: ${\lambda }_1,S,D$ , tolerance $ϵ$

Initialize: $L^{\left(0\right)}=D$

Repeat:

Step 1. Given $L^{\left(k\right)}$ , $\left[U^{\left(k\right)},{\Sigma }^{\left(k\right)},V^{\left(k\right)}\right]=svd\left(L^{\left(k\right)}\right)$ , then let $A^{\left(k\right)}$ , $B^{\left(k\right)}$ being the first r columns’ transpose of $U^{\left(k\right)}$ and $V^{\left(k\right)}$ respectively.

Step 2, solve

$L^{\left(k+1\right)}=\arg \underset{L}{\min }\frac{1}{2}{\Vert D-L-S\Vert }_F^2+{\lambda }_1\left({\Vert L\Vert }_{\ast }-Tr\left(A^{\left(k\right)}L{B}^{\left(k\right)}{}^{{}^{\text{T}}}\right)\right)$ (14)

Until ${\Vert L^{\left(k+1\right)}-L^{\left(k\right)}\Vert }_F\le ϵ$

Output $L^{\left(k+1\right)}$