Algorithm 2 Inexact ALM method for solving RPCA.

Input:  Observation matrix $\mathbf{X}\in {\mathbb{R}}^{m\times n}$ and parameter $\lambda$ for local affinity.   1:Initialize: ${\mathbf{Y}}_0=\mathbf{X}/J\left(\mathbf{X}\right);{\mathbf{E}}_0=\mathbf{0};{\mu }_0>0;\rho >1;k=0$.   2:while not converged do   3:    //Lines 4-5 resolve ${\mathbf{Z}}_{i+1}=\underset{\mathbf{Z}}{\arg \min }L(\mathbf{Z},{\mathbf{E}}_i,{\mathbf{Y}}_i,{\mu }_i)$.   4:    $(\mathbf{U},\mathbf{S},\mathbf{V})=svd(\mathbf{X}-{\mathbf{E}}_i+{{\mu }_i}^{-1}{\mathbf{Y}}_i)$;   5:    ${\mathbf{Z}}_{i+1}=\mathbf{U}{\mathbf{S}}_{{{\mu }_i}^{-1}}\left[\mathbf{S}\right]{\mathbf{V}}^T$.   6:    //Line 7 resolves ${\mathbf{E}}_{i+1}=\underset{\mathbf{E}}{\arg \min }L({\mathbf{Z}}_{i+1},\mathbf{E},{\mathbf{Y}}_i,{\mu }_i)$.   7:    ${\mathbf{E}}_{i+1}={\mathbf{S}}_{{{\mu }_i}^{-1}}(\mathbf{X}-{\mathbf{Z}}_{i+1}+{{\mu }_i}^{-1}{\mathbf{Y}}_i)$.   8:    ${\mathbf{Y}}_{i+1}={\mathbf{Y}}_i+{\mu }_i(\mathbf{X}-{{\mathbf{Z}}_{i+1}}^{*}-{{\mathbf{E}}_{i+1}}^{*});{\mu }_{i+1}=\rho {\mu }_i$.   9:    $i\leftarrow i+1$.  10: end while Output:  The low-rank matrix $\mathbf{Z}$.