# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 4 Nonconvex Nonlinear Transformation for Tensor Completion
### Optimization Using LRANNT-PAM
 \tag{10}\]  
where \(i=1,2,\ldots,t_{n}\). Following Equation (10), \(\mathbf{G}_{n}\) is expressed the following relaxed form:  
\[ \begin{split}&\arg\min_{\mathbf{G}_{n}}\frac{1}{\hat{\rho}_{n}} \sum_{n=1}^{t_{n}}\varphi\Big{(}\sigma_{n}^{k}\Big{)}+\
abla\varphi\Big{(} \sigma_{n}^{k}\Big{)}\Big{(}\sigma_{n}(\mathbf{G}_{n})-\sigma_{n}^{k}\Big{)}+ f(\mathbf{G}_{n})\\ &=\arg\min_{\mathbf{G}_{n}}\frac{1}{\hat{\rho}_{n}}\sum_{n=1}^{t _{n}}\
abla\varphi\Big{(}\sigma_{n}^{k}\Big{)}\sigma_{n}(\mathbf{G}_{n})+ \frac{1}{2}\Big{\|}\mathbf{G}_{n}-\mathbf{N}_{n}^{k}\Big{\|}_{F}^{2}.\end{split}  \tag{11}\]  
Regarding nonconvex nonsmooth low-rank minimization problems [36; 38], the solution to Equation (11) can be effectively derived through the utilization of the singular value thresholding (SVT) operator [41]. The globally optimal solution \(\mathbf{G}_{n}^{k+1}\) is given as follows:  
\[\mathbf{G}_{n}^{k+1}=\text{SVT}\bigg{(}\mathbf{N}_{n}^{k},\frac{\
abla\varphi }{\hat{\rho}_{n}}\bigg{)}=\mathcal{S}_{\frac{\
abla\varphi}{\hat{\rho}_{n}}} \Big{(}\mathbf{H}_{i}^{k}\Big{)}=\mathbf{U}\mathcal{S}_{\frac{\
abla\varphi} {\hat{\rho}_{n}}}(\mathbf{D})\mathbf{V}^{T}, \tag{12}\]  
where \(\mathbf{N}_{n}^{k}=\mathbf{U}\mathbf{D}\mathbf{V}^{T}\) is the SVD of \(\mathbf{N}_{n}^{k}\),  
\[\mathcal{S}_{\frac{\
abla\varphi}{\hat{\rho}_{n}}}(\mathbf{D})=\text{diag} \left(\max\left(\Sigma_{j}-\frac{\
abla\varphi\Big{(}\mathbf{e}_{i}^{k}\Big{)} }{\hat{\rho}_{n}},0\right)\right), \tag{13}\]  
and \(i=1,2,\ldots,t_{n}\). \(\bullet\)**Updating**\(\mathcal{X}\)  
Referring to Equation (5), the update process of \(\mathcal{X}\) can be simplified as:  
\[ \begin{split}\mathcal{X}^{k+1}&\in\underset{ \mathcal{X}}{\operatorname{argmin}}\bigg{\{}\mathcal{L}\Big{(}\mathcal{G}^{k+1},\mathcal{X},\mathcal{H}^{k},\mathbf{M}^{k}\Big{)}+\frac{\rho_{2}}{2}\Big{\|} \mathcal{X}-\mathcal{X}^{k}\Big{\|}_{F}^{2}\bigg{\}},\\ &=\underset{\mathcal{X}}{\operatorname{argmin}}\frac{\alpha}{2} \Big{\|}\mathcal{X}-\mathcal{H}^{k}\times_{3}\mathbf{M}^{k^{\top}}\Big{\|}_ {F}^{2}+\frac{\rho_{2}}{2}\Big{\|}\mathcal{X}-\mathcal{X}^{k}\Big{\|}_{F}^{2}+ \Gamma(\mathcal{X}),\end{split}  \tag{14}\]where \(\rho_{2}>0\). Expanding the terms and manipulating the expression, we obtain  
\[ \begin{split}\underset{\mathcal{X}}{\operatorname{argmin}}\frac{ \alpha}{2}\|\mathcal{X}\|_{F}^{2}-\alpha\langle\mathcal{X},\mathcal{H}^{k} \times_{3}\mathbf{M}^{k^{\top}}\rangle+\frac{\alpha}{2}\Big{\|}\mathcal{H}^{k }\times_{3}\mathbf{M}^{k^{\top}}\Big{\|}_{F}^{2}\\ +\frac{\rho_{2}}{2}\|\mathcal{X}\|_{F}^{2}-\rho_{2}\langle\mathcal{ X},\mathcal{X}^{k}\rangle+\frac{\rho_{2}}{2}\Big{\|}\mathcal{X}^{k}\Big{\|}_{F}^{2}+ \Gamma(\mathcal{X}).\end{split}  \tag{15}\]  
By taking the derivative with respect to \(\mathcal{X}\), we derive the following expression:  
\[\alpha\mathcal{X}+\rho_{2}\mathcal{X}-\alpha\mathcal{H}^{k}\times_{3}\mathbf{ M}^{k^{\top}}-\rho_{2}\mathcal{X}^{k}+\
abla\Gamma(\mathcal{X})=0, \tag{16}\]  
where the sum of the gradients of the objective terms vanishes. Solving for \(\mathcal{X}\), we obtain  
\[\mathcal{X}^{k+1}=\left(\frac{\alpha\mathcal{H}^{k}\times_{3}\mathbf{M}^{k^{ \top}}+\rho_{2}\mathcal{X}^{k}}{\alpha+\rho_{2}}\right)_{\Omega^{\times}}+ \mathcal{O}_{\Omega}, \tag{17}\]  
where \(\
abla\varphi(\mathcal{X})\) denotes the gradient of the indicator function \(\varphi(\mathcal{X})\), and \(\Omega^{\times}\) represents the complement set of \(\Omega\). \(\bullet\)**Updating \(\mathcal{H}\)**  
The update process of \(\mathcal{H}^{k+1}\) can be simplified as:  
\[