Algorithm 2. Solve Equation (10) by ADMM

Input: tensor $\mathcal{Z}\in {\mathbb{R}}^{n_1\times n_2\times n_3}$, parameters $\lambda ,\tau ,\gamma ,\mu ,\xi$, maximum iterations $N$;

While convergence is not satisfied do

1. Update ${\mathcal{X}}_{k+1}$ by: ${\mathcal{X}}_{k+1}=\underset{\mathcal{X}}{\arg \min }{\displaystyle \sum _{i=1}^{n_3}{\displaystyle \sum _{j=1}^n{f}_{\tau ,\gamma }({\sigma }_j^{(i)}(\hat{\mathcal{X}}))}}+\frac{\mu }{2}{\Vert \mathcal{X}-(-{\mathcal{G}}_k+\mathcal{Z}-{\mathcal{M}}_k)\Vert }_F^2$; 2. Update ${\mathcal{G}}_{k+1}$ by: ${\mathcal{G}}_{k+1}=\underset{\mathcal{G}}{\arg \min } \lambda {\Vert \mathcal{G}\Vert }_1+\frac{\mu }{2}{\Vert \mathcal{G}-(-{\mathcal{X}}_{k+1}+\mathcal{Z}-{\mathcal{M}}_k)\Vert }_F^2$;

3. Update ${\mathcal{M}}_{k+1}$ by: ${\mathcal{M}}_{k+1}={\mathcal{M}}_k+{\mathcal{X}}_{k+1}+{\mathcal{G}}_{k+1}-\mathcal{Z}$;

4. Check for convergence: ${\Vert {\mathcal{X}}_{k+1}-{\mathcal{X}}_k\Vert }_{\infty }\le \xi ,{\Vert {\mathcal{G}}_{k+1}-{\mathcal{G}}_k\Vert }_{\infty }\le \xi ,{\Vert {\mathcal{X}}_{k+1}+{\mathcal{G}}_{k+1}-\mathcal{Z}\Vert }_{\infty }\le \xi$

End

Output: $\mathcal{X}$