# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 4 Nonconvex Nonlinear Transformation for Tensor Completion
### Optimization of Parallel Computing  
Given that solving a large number of SVD decompositions in Equation (12) will lead to an increase in time complexity, we have proposed a parallel computing optimized model based on LRANNT. This model applies parallel matrix decomposition to the third mode, aiming to reduce computation time of SVD. We start by decomposing \(\mathbf{G}_{(3)}\) using parallel low-rank matrix decomposition:  
\[\mathbf{G}_{(3)}=\mathbf{AB}, \tag{26}\]  
where, \(\mathbf{A}\in\mathbb{R}^{I\times d}\), \(\mathbf{B}\in\mathbb{R}^{d\times\Pi_{j\
eq n}I_{j}}\), \(d_{n}\geq\operatorname{rank}(\mathbf{G}_{(3)})\), and \(d_{n}\ll\min\{I_{n},\Pi_{j\
eq n}I_{j}\}\). Then, we impose rank constraints on the factor matrices \(\mathbf{A}\) and \(\mathbf{B}\):  
\[\arg\underset{\mathbf{G}_{(3)}=\mathbf{AB}}{\operatorname{min}}(\tau \operatorname{rank}(\mathbf{A})+\lambda\operatorname{rank}(\mathbf{B})), \tag{27}\]  
where \(\tau\) and \(\lambda\) are non-negative regularization parameters that control the rank of \(\mathbf{A}\) and \(\mathbf{B}\), respectively. The optimized LRANNT model is formulated as:  
\[ \begin{split}\min_{\mathcal{L}(\mathbf{A},\mathbf{B},\mathcal{X}, \mathcal{H},\mathbf{M})}&\sum_{i=1}^{r}(\tau\|\mathbf{A}\|_{ \gamma}+\lambda\|\mathbf{B}\|_{\gamma}+\frac{\theta}{2}\|\mathbf{G}_{(3)}- \mathbf{A}\mathbf{B}\|_{\mathrm{F}}^{2})\\ &\quad+\frac{\alpha}{2}\|\mathcal{X}-\mathcal{Z}\times_{3} \mathbf{M}^{\top}\|_{\mathrm{F}}^{2}+\frac{\beta}{2}\|\mathcal{Y}-\varphi( \mathcal{Z})\|_{\mathrm{F}}^{2}+\varphi(\mathcal{X})+\delta(\mathbf{M}).\end{split}  \tag{28}\]  
Next, since the updates of variables \(\mathcal{X}\), \(\mathcal{H}\), and \(\mathbf{M}\) are consistent with Algorithm 1, we omit the derivation of this part and only derive the updates of variables \(\mathbf{A}\) and \(\mathbf{B}\) after the decomposition of \(\mathbf{G}_{(3)}\). \(\bullet\)**Updating A**  
The optimization subproblem in (28) with respect to \(\mathbf{A}\) is expressed as follows after other variables have been fixed:  
\[\underset{\mathbf{A}}{\operatorname{argmin}}\frac{\theta}{2}\Big{\|}\mathbf{ G}_{(3)}-\mathbf{A}\mathbf{B}\Big{\|}_{\mathrm{F}}^{2}+\tau\|\mathbf{A}\|_{ \gamma}+\frac{\rho}{2}\Big{\|}\mathbf{A}-\mathbf{A}^{k}\Big{\|}_{\mathrm{F}}^ {2}. \tag{29}\]  
Let \(\mathbf{A}=\mathbf{J}\), and the augmented Lagrangian multiplier (ALM) of Equation (29) can be expressed as:  
\[\underset{\mathbf{A}\mathbf{J}}{\operatorname{argmin}}\frac{\theta}{2}\Big{\|} \mathbf{G}_{(3)}-\mathbf{A}\mathbf{B}\Big{\|}_{\mathrm{F}}^{2}+\frac{\rho}{2 }\Big{\|}\mathbf{A}-\mathbf{A}^{k}\Big{\|}_{\mathrm{F}}^{2}+\Big{\langle} \Gamma^{\mathbf{A}},\mathbf{A}-\mathbf{J}\Big{\rangle}+\tau\|\mathbf{J}\|_{ \gamma}+\frac{\rho}{2}\|\mathbf{A}-\mathbf{J}\|_{\mathrm{F}}^{2}, \tag{30}\]  
where \(\Gamma^{\mathbf{A}}\) represents Lagrangian multipliers. By fixing the other parameters, \(\mathbf{J}\) is updated according to Equation (30):  
\[\mathbf{J}^{k+1}=\arg\underset{\mathbf{J}}{\operatorname{min}}\|\mathbf{J}\|_ {\gamma}+\frac{\hat{\rho}}{2}\Big{\|}\mathbf{J}-\mathbf{P}^{k}\Big{\|}_{ \mathrm{F}}^{2}, \tag{31}\]  
where \(\hat{\rho}=\rho/\tau,\mathbf{P}^{k}=\mathbf{A}^{k}+\Gamma^{\mathbf{A}}/\rho\). Its solution can be obtained by Lemma 1 as follows:  
\[\mathbf{J}^{k+1}=\mathrm{SVT}\bigg{(}\mathbf{P}^{k},\frac{\
abla\varphi}{ \rho}\bigg{)}=\mathbf{U}\mathcal{S}_{\frac{\
abla\varphi}{\rho}}(\mathbf{D}) \mathbf{V}^{\mathrm{T}}, \tag{32}\]  
where \(\mathbf{P}^{k}=\mathbf{U}\mathbf{D}\mathbf{V}^{\mathrm{T}}\) is the SVD of \(\mathbf{P}^{k}\),  
\[\mathbf{S}_{\frac{\
abla\varphi}{\rho}}(\mathbf{D})=\text{Diag}\bigg{(}\max \bigg{(}\mathbf{\Sigma}_{ii}-\frac{\
abla\varphi(\mathbf{\sigma}_{i}^{k})}{\beta_{ n}},0\bigg{)}\bigg{)}, \tag{33}\]  
and \(i=1,2,\ldots,t_{n}\).