# On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery
## 3 Proposed Method
```
0:\(\mathcal{Y},\sigma_{n}\), \(\boldsymbol{\gamma}=[\gamma_{1},\cdots,\gamma_{N}]\), \(\mathbf{w}=[\mathbf{w}_{1}^{\top},\cdots,\mathbf{w}_{N}^{\top}]^{\top}\), \(p\), \(\lambda\)
1: Initialize \(\mathbf{Y}_{1,m}^{(0)}=\text{unfold}_{m}(\mathcal{Y})\), \(\mathcal{Y}_{2}^{(0)}=\mathcal{Y}\), \(\mathbf{Z}_{1,m}^{(0)}=0\), \(\mathbf{Y}_{1,m}^{(0)}\), \(\mathcal{Z}_{2}^{(0)}=0\)
2:while A stopping criterion is not satisfieddo
3:\(\mathcal{X}^{(k+1)}=\underset{\mathcal{X}}{\text{argmin}}\ \frac{1}{2}\sum_{m=1}^{N}\|\mathbf{Y}_{1,m}^{(k)}-\text{unfold}_{m}( \mathcal{X})-\mathbf{Z}_{1,m}^{(k)}\|_{2}^{2}+\lambda\|\mathcal{Y}_{2}^{(k)}- \mathcal{Z}_{2}^{(k)}\|_{2}^{2}\)
4:for\(m=1\) to \(N\)do
5:\(\mathbf{Y}_{1,m}^{(k+1)}=\underset{\mathbf{X}}{\text{argmin}}\ \|\mathbf{X}\|_{\lambda\gamma_{m}\mathbf{w}_{m},p}^{p}+\frac{1}{2}\|\text{ unfold}_{m}(\mathcal{X}^{(k+1)})+\mathbf{Z}_{1,m}^{(k)}-\mathbf{X}\|_{\text{F}}^{2}\)
6:\(\mathbf{Z}_{1,m}^{(k+1)}=\mathbf{Z}_{1,m}^{(k)}+\text{unfold}_{m}(\mathcal{X} ^{(k+1)})-\mathbf{Y}_{1,m}^{(k+1)}\)
7:endfor
8:\(\mathcal{Y}_{2}^{(k+1)}=\text{proj}_{\text{B}(\mathcal{Y},\sigma_{n}\sqrt{| \alpha|})}(\Lambda_{\Omega}(\mathcal{X}^{(k+1)}+\mathcal{Z}_{2}^{(k)}))+ \Lambda_{\Omega}(\mathcal{X}^{(k+1)}+\mathcal{Z}_{2}^{(k)})\)
9:\(\mathcal{Z}_{2}^{(k+1)}=\mathcal{Z}_{2}^{(k)}+\Lambda_{\Omega}(\mathcal{X}^{ (k+1)})-\mathcal{Y}_{2}^{(k+1)}\)
10:\(\lambda=0.99\lambda\)
11:\(k=k+1\)
12:endwhile
13:\(\mathcal{X}^{(k)}\)
```  
**Algorithm 1** Proposed algorithm  
Each element of the weighted thresholding operator for a rectangular diagonal matrix \(Y\ (\mathbf{S}_{\mathbf{w},p}(\mathbf{Y}))_{i,j}\) is defined as a solution to the following minimization problem:  
\[\underset{\mathbf{x}}{\text{argmin}}\ \frac{1}{2}(\mathbf{Y}_{i,i}-\mathbf{x})^{2}+ \mathbf{w}_{i}|\mathbf{x}|^{p}. \tag{13}\]  
The solution of Equation (13) is a soft thresholding \(\max(\mathbf{Y}_{i,i}-\mathbf{w}_{i},0)\) when \(p=1\) and the closed-form thresholding proposed in [28] when \(p=\{1/2,2/3\}\). The first term in line 8 of Algorithm 1 is  
\[\text{proj}_{\text{B}(\mathcal{Y},\sigma_{n}\sqrt{|\Omega|})}(\Lambda_{\Omega }(\mathcal{X}^{(k+1)}+\mathcal{Z}_{2}^{(k)})) \tag{14}\]  
which is a metric projection of the set \(\text{B}(\mathcal{Y},\sigma_{n}\sqrt{|\Omega|})\). The metric projection is defined as  
\[\text{proj}_{\text{S}}\colon\mathbb{R}^{N}\to\mathbb{R}^{N}\colon\mathbf{x} \mapsto\underset{\mathbf{y}\in S}{\text{argmin}}\ \frac{1}{2}\|\mathbf{x}-\mathbf{y}\|_{2}^{2}. \tag{15}\]  
Equation (14) has a closed-form solution,\[\mathcal{Y}-\min\Bigg{(}\frac{\sigma_{n}\sqrt{|\Omega|}}{\|\mathcal{Y}-A_{\Omega}( \mathcal{X}^{(k+1)}+\mathcal{Z}_{2}^{(k)})\|_{2}},1\Bigg{)}(\mathcal{Y}-A_{ \Omega}(\mathcal{X}^{(k+1)}+\mathcal{Z}_{2}^{(k)})). \tag{16}\]  
The set of \(\tilde{\Omega}\) in the second term of line 8,  
\[A_{\Omega}(\mathcal{X}^{(k+1)}+\mathcal{Z}_{2}^{(k)}) \tag{17}\]  
is the complement of the set \(\Omega\), which is a set of indicators of missing entries. \(A_{\Omega}\) is defined as  
\[A_{\tilde{\Omega}}(\mathcal{X})_{i_{1},\cdots,i_{N}}=\left\{ \begin{array}{ ll}\mathcal{X}_{i_{1},\cdots,i_{N}}&(i_{1},\cdots,i_{N}\in\tilde{\Omega})\\ 0&(\text{otherwise})\end{array} \right.. \tag{18}\]