# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 4 Nonconvex Nonlinear Transformation for Tensor Completion
### Optimization Using LRANNT-PAM
\begin{split}\mathcal{H}^{k+1}&\in\underset{ \mathcal{H}}{\operatorname{argmin}}\bigg{\{}\mathcal{L}\Big{(}\mathcal{Q}^{k+1 },\mathcal{X}^{k+1},\mathcal{H},\mathbf{M}^{k}\Big{)}+\frac{\rho_{3}}{2}\Big{\|} \mathcal{H}-\mathcal{H}^{k}\Big{\|}_{F}^{2}\bigg{\}},\\ &=\underset{\mathcal{H}}{\operatorname{argmin}}\frac{\alpha}{2} \Big{\|}\mathcal{X}^{k+1}-\mathcal{H}\times_{3}\mathbf{M}^{k^{\top}}\Big{\|} _{F}^{2}+\frac{\beta}{2}\Big{\|}\mathcal{Q}^{k+1}-\varphi(\mathcal{H})\Big{\|} _{F}^{2}+\frac{\rho_{3}}{2}\Big{\|}\mathcal{H}-\mathcal{H}^{k}\Big{\|}_{F}^{2 },\end{split}  \tag{18}\]  
where \(\rho_{3}>0\). This subproblem is equivalently expressed in terms of the matricized third mode \(\mathbf{H}_{(3)}\) as:  
\[\underset{\mathbf{H}_{(3)}}{\operatorname{argmin}}\frac{\alpha}{2}\Big{\|} \mathbf{H}_{(3)}-\mathbf{M}^{k}\mathbf{X}_{(3)}^{k+1}\Big{\|}_{F}^{2}+\frac{ \beta}{2}\Big{\|}\varphi\Big{(}\mathbf{H}_{(3)}\Big{)}-\mathbf{G}_{(3)}^{k+1} \Big{\|}_{F}^{2}+\frac{\rho_{3}}{2}\Big{\|}\mathbf{H}_{(3)}-\mathbf{H}_{(3)}^{ k}\Big{\|}_{F}^{2}. \tag{19}\]  
Setting \(\mathbf{Q}^{k}=\frac{\alpha\mathbf{M}^{k}\mathbf{X}_{(3)}^{k+1}+\rho_{3} \mathbf{H}_{(3)}^{k}}{\alpha+\rho_{3}}\). The optimization problem (19) is addressed as follows:  
\[h_{ij}^{k+1}=\frac{\alpha+\rho_{3}}{2}\Big{(}h_{ij}-q_{ij}^{k}\Big{)}^{2}+\frac {\beta}{2}\Big{(}\varphi(h_{ij})-g_{ij}^{k+1}\Big{)}^{2}. \tag{20}\]  
where the \((i,j)\)-th element of the matrices \(\mathbf{H}_{(3)}^{k}\), \(\mathbf{Q}^{k}\), and \(\mathbf{G}_{(3)}^{k}\) is denoted as \(h_{ij}^{k}=\mathbf{H}_{(3)}^{k}(i,j)\), \(q_{ij}^{k}=\mathbf{Q}^{k}(i,j)\), and \(g_{ij}^{k}=\mathbf{G}_{(3)}^{k+1}(i,j)\), where the update of \(\mathbf{H}_{(3)}\) in Equation (19) is to be decomposed into \(n_{1}n_{2}r\) such linear Equations [27]. \(\bullet\)**Updating \(\mathbf{M}\)**  
The update process of \(\mathbf{M}^{k+1}\) can be simplified as:  
\[ \begin{split}\mathbf{M}^{k+1}&\in\underset{ \mathbf{M}}{\operatorname{argmin}}\bigg{\{}\mathcal{L}\Big{(}\mathcal{Q}^{k+1 },\mathcal{X}^{k+1},\mathcal{H}^{k+1},\mathbf{M}\Big{)}+\frac{\rho_{4}}{2} \Big{\|}\mathbf{M}-\mathbf{M}^{k}\Big{\|}_{F}^{2}\bigg{\}},\\ &=\underset{\mathbf{M}}{\operatorname{argmin}}\frac{\alpha}{2} \Big{\|}\mathcal{X}^{k+1}-\mathcal{H}^{k+1}\times_{3}\mathbf{M}^{\top}\Big{\|} _{F}^{2}+\frac{\rho_{4}}{2}\Big{\|}\mathbf{M}-\mathbf{M}^{k}\Big{\|}_{F}^{2}+ \delta(\mathbf{M}),\end{split}  \tag{21}\]  
where \(\rho_{4}>0\). Consider the transformation of problem (21) in terms of the matricized third mode into the following equivalent form:  
\[\underset{\mathbf{M}}{\operatorname{argmin}}\frac{\alpha}{2}\Big{\|}\mathbf{X} _{(3)}^{k+1}-\mathbf{M}^{\top}\mathbf{H}_{(3)}^{k+1}\Big{\|}_{F}^{2}+\frac{ \rho_{4}}{2}\Big{\|}\mathbf{M}-\mathbf{M}^{k}\Big{\|}_{F}^{2}+\delta(\mathbf{M}). \tag{22}\]By expanding the matrix multiplication and transposition inside the trace function, we can obtain the following expanded form:  
\[ \begin{split}\underset{\mathbf{M}}{\operatorname{argmin}}& \frac{\alpha}{2}\operatorname{Tr}\left[\left(\mathbf{X}_{(3)}^{k+1}- \mathbf{M}^{\top}\mathbf{H}_{(3)}^{k+1}\right)^{\top}\left(\mathbf{X}_{(3)}^{ k+1}-\mathbf{M}^{\top}\mathbf{H}_{(3)}^{k+1}\right)\right]\\ &+\frac{\rho_{4}}{2}\operatorname{Tr}\left[\left(\mathbf{M}- \mathbf{M}^{k}\right)^{\top}\left(\mathbf{M}-\mathbf{M}^{k}\right)\right]+ \delta(\mathbf{M}).\end{split}  \tag{23}\]  
Taking the negative of this quadratic form, it is transformed into a maximization problem:  
\[\underset{\mathbf{M}}{\operatorname{argmax}}\operatorname{Tr}\left[\left( \alpha\mathbf{X}_{(3)}^{k+1}\left(\mathbf{H}_{(3)}^{k+1}\right)^{\top}+\rho_{4} \mathbf{M}^{k^{\top}}\right)\mathbf{M}\right]-\delta(\mathbf{M}), \tag{24}\]  
where \(\operatorname{Tr}(\mathbf{X})\) represents the trace of matrix \(\mathbf{X}\). Suppose the SVD of \(\alpha\mathbf{X}_{(3)}^{k+1}\left(\mathbf{H}_{(3)}^{k+1}\right)^{\top}+\rho_{ 4}\mathbf{M}^{k^{\top}}\) is \(\mathbf{UDV}^{\top}\).