Algorithm 1. Proposed Robust ${\ell }_{2,1}$-norm Regularized Matrix Decomposition (RRMD) Algorithm

Input: The sampled EDM matrix $\mathit{M}$, the index set $\mathsf{\Omega }$, the parameters ${\lambda }_1$, ${\lambda }_2$ and threshold $\theta$, the initial number of Gaussian components $n_c$.

Output: The true underlying EDM matrix $\widehat{\mathit{D}}$, the row structural anomaly $\mathit{R},$ and the column structural anomaly $\mathit{C}$.

1. Randomly initialize ${\mathit{U}}_0,{\mathit{V}}_0,{\Pi }_0,{\Sigma }_0$; initialize ${\mathit{R}}_0,{\mathit{C}}_0$ to zero matrix;

2. $t=1$;

3. While not convergence do

4.  (E-step): update $\mathit{{\rm Y}}$ according to Equation (17);

5.  (M-step): update $\Pi ,\Sigma$ according to Equation (19) and Equation (20);

6.  (M-step): update $\mathit{U},\mathit{V}$ according to optimize Equation (24);

7.  (M-step): update $\mathit{R}$ according to Equation (25);

8.  (M-step): update $\mathit{C}$ according to Equation (26);

9.  (Tuning $n_c$): Let $g_i$ and $g_i$ represent the number of $i$-th and$j$-th Gaussian component respectively. if $|{\sigma }_i{}^2-{\sigma }_j{}^2|/\left({\sigma }_i{}^2+{\sigma }_j{}^2\right)<\theta$, then let ${\pi }_i={\pi }_i+{\pi }_j$, ${\sigma }_i{}^2=\left(g_i{\sigma }_i{}^2+g_j{\sigma }_j{}^2)/(g_i+g_j\right)$, $K=K-1$. Lastly, remove ${\pi }_j$ and ${\sigma }_j^2$ from $\Pi ,\Sigma$, respectively.

10.  $t=t+1$;

11. End while

12. $\mathit{R},\mathit{C} \mathrm{and} \widehat{\mathit{D}}=\mathit{U}{\mathit{V}}^T$.