# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 4 Nonconvex Nonlinear Transformation for Tensor Completion
### Optimization Using LRANNT-PAM  
In this framework, the optimization problem in Equation (3) can be reformulated as follows:  
\[\min_{\mathcal{X},\mathcal{H},\mathbf{M}}\sum_{i=1}^{r}\lVert\varphi(\mathcal{ H}_{i})\rVert_{\gamma}+\Gamma(\mathcal{X})+\delta(\mathbf{M}),\text{ s.t. }\mathcal{X}=\mathcal{H}\times_{3}\mathbf{M}^{\top}, \tag{4}\]  
where \(\Gamma(\mathcal{X})\) and \(\delta(\mathbf{M})\) are indicator functions. \(\Gamma(\mathcal{X})=0\) if the projection of \(\mathcal{X}\) at specified positions matches a fixed matrix \(\mathcal{O}\), and \(\infty\) otherwise. Similarly, \(\delta(\mathbf{M})=0\) if \(\mathbf{M}\mathbf{M}^{\top}=I\), and \(\infty\) otherwise. These functions impose constraints on the optimization problem. Setting \(\mathcal{G}=\varphi(\mathcal{H})\) and \(\mathcal{X}=\mathcal{H}\times_{3}\mathbf{M}^{\top}\) using the semi-definite splitting technique [40] on another constraint, problem (4) is transformed as follows:  
\[\min_{\mathcal{G},\mathcal{X},\mathcal{H},\mathbf{M}}\mathcal{L} (\mathcal{G},\mathcal{X},\mathcal{H},\mathbf{M})=\min_{\mathcal{G},\mathcal{X}, \mathcal{H},\mathbf{M}}\sum_{i=1}^{r}\lVert\mathbf{G}_{i}\rVert_{\gamma} +\frac{\alpha}{2}\left\lVert\mathcal{X}-\mathcal{H}\times_{3} \mathbf{M}^{\top}\right\rVert_{F}^{2} \tag{5}\] \[+\frac{\beta}{2}\lVert\mathcal{G}-\varphi(\mathcal{H})\rVert_{F} ^{2}+\Gamma(\mathcal{X})+\delta(\mathbf{M}),\]  
where \(\alpha,\beta>0\) are two regularization penalty parameters. \(\bullet\)**Updating**\(\mathcal{G}\)  
During the update of \(\mathcal{G}\), the objective function is \(f(\mathcal{G})\). Within the framework of the PAM algorithm, we let \(K\) represent the number of iterations. According to Equation (5), the update process can be simplified as:  
\[\mathcal{G}^{k+1} \in\operatorname*{argmin}_{\mathcal{G}}\left\{\mathcal{L}\left( \mathcal{G},\mathcal{X}^{k},\mathcal{H}^{k},\mathbf{M}^{k}\right)+\frac{\rho_{ 1}}{2}\left\lVert\mathcal{G}-\mathcal{G}^{k}\right\rVert_{F}^{2}\right\}, \tag{6}\] \[=\operatorname*{argmin}_{\mathcal{G}}\sum_{i=1}^{r}\lVert \mathbf{G}_{i}\rVert_{\gamma}+\frac{\beta}{2}\left\lVert\mathcal{G}-\varphi \left(\mathcal{H}^{k}\right)\right\rVert_{F}^{2}+\frac{\rho_{1}}{2}\left\lVert \mathcal{G}-\mathcal{G}^{k}\right\rVert_{F}^{2},\]where \(\rho_{1}>0\). The \(\mathcal{G}\) subproblem is dissected into a set of subproblems, expressed as follows:  
\[\underset{\mathbf{G}_{i}}{\operatorname{argmin}}\|\mathbf{G}_{i}\|_{\gamma}+ \frac{\beta}{2}\Big{\|}\mathbf{G}-\varphi\Big{(}\mathbf{H}^{k}\Big{)}\Big{\|} _{F}^{2}+\frac{\rho_{1}}{2}\Big{\|}\mathbf{G}-\mathbf{G}^{k}\Big{\|}_{F}^{2}, \tag{7}\]  
where the optimization problem for \(\mathbf{G}_{i}\) is inferred from (7) as:  
\[\mathbf{G}^{k+1}=\underset{\mathbf{G}_{i}}{\operatorname{argmin}}\|\mathbf{G }_{i}\|_{\gamma}+\frac{\hat{\rho}}{2}\|\mathbf{G}_{i}-\mathbf{N}_{i}\|_{F}^{2}, \tag{8}\]  
where \(\mathbf{N}_{i}^{k}=\frac{\beta\varphi(\mathbf{H}_{i}^{k})+\rho_{1}\mathbf{G}_ {i}^{k}}{\beta+\rho_{1}}\), \(\hat{\rho}=\beta+\rho_{1}\). Let \(\sigma_{1}^{k}\geq\sigma_{2}^{k}\geq\cdots\geq\sigma_{t_{n}}^{k}\) represent the singular values of \(\mathbf{G}_{n}^{k}\in\mathbb{R}^{r_{n}\times s_{n}}\) and \(t_{n}=\min\{r_{n},s_{n}\}\). \(\
abla\varphi\Big{(}\sigma_{n}^{k}\Big{)}\) represents the gradient of \(\varphi(x)=1-e^{-x/\gamma}\) at \(\sigma_{n}^{k}\). Define  
\[f(\mathbf{G}_{n})=\frac{1}{2}\Big{\|}\mathbf{G}_{n}-\mathbf{N}_{n}^{k}\Big{\|} _{F}^{2}, \tag{9}\]  
where, when the Lipschitz constant is \(1\), \(f(\mathbf{G}_{n})\) has a Lipschitz continuous gradient. Therefore, based on the antimonotone property of the gradient of the nonconvex function and the nonascending order of singular values, we conclude that:  
\[ \begin{split}& 0\leq\
abla\varphi\Big{(}\sigma_{1}^{k}\Big{)} \leq\
abla\varphi\Big{(}\sigma_{2}^{k}\Big{)}\leq\cdots\leq\
abla\varphi \Big{(}\sigma_{n}^{k}\Big{)},\\ &\varphi(\sigma_{i}(\mathbf{G}_{n}))\leq\varphi\Big{(}\sigma_{i} ^{k}\Big{)}+\
abla\varphi\Big{(}\sigma_{i}^{k}\Big{)}\Big{(}\sigma_{i}( \mathbf{G}_{n})-\sigma_{i}^{k}\Big{)},\end{split}