## 4 The Non-Convex Approximate RPCA Algorithm
### Derivation of the Overall Formula for NCA-RPCA Algorithm  
From Sections 4.1 and 4.2, the optimization problem can be redefined as follows:  
\[\min_{L,S}\ \sum_{j}\frac{(1+\gamma)\sigma_{j}(\mathbf{L})}{\gamma+\sigma_{j}(\mathbf{L})}+ \sum_{ij}f_{\lambda}(s_{ij})\ \ \ \text{s.t.}\mathbf{M}=\mathbf{L}+\mathbf{S} \tag{18}\]  
where \(s_{ij}\) represents the element in the \(i\)th row and \(j\)th column of matrix \(S\), the parameter \(\lambda\) determines the relative impact of low-rank and sparsity on the objective function. The optimal value of \(\lambda\) that balances the minimization of the rank of \(L\) and the maximization of the sparsity of \(S\) can be achieved by controlling \(\lambda\). To determine the parameter \(\lambda\), numerical experiments were conducted in this study, using the number of iterations and error value (err) as criteria for selection. Let \(\lambda=ini/\sqrt{\max(m,n)}\), where \(ini\) is set within the range \([0.1,10]\), and m and n are the rows and columns of the input image matrix, respectively. Figure 6 shows the relationship between the number of iterations, error value (err), and \(\lambda\). From Figure 6, it can be observed that as \(ini\) increases, the number of iterations continuously rises, and the error value also increases to some extent. Therefore, based on Figure 5, this study chooses \(ini=0.2\) to determine the value of parameter \(\lambda\). The role of \(\lambda\) is to balance the low-rank and sparse components of the matrix, while \(\rho\) affects the threshold of the non-convex penalty function. Thus, these two parameters need to be jointly adjusted to achieve a balance between the low-rank and sparse components. The penalty coefficient \(\rho\), following the approach in reference [30], is set as \(\rho=4\lambda\). This process ultimately leads to a well-optimized set of parameters for the algorithm. Figure 5: Super-gradients function. Figure 6: Illustrates the relationship curve between the number of iterations, error value, and \(ini\). When seeking a solution for Equation (18), the properties of the function can be derived from the gradient (or super-gradient when there is a non-smooth origin point). At the point \(\alpha\in R^{n}\), if \(f(\alpha)+\left\langle u,\beta-\alpha\right\rangle\geq f(\beta)\) holds for every \(\beta\in R^{n}\); then, \(u\) is the super-gradient of function \(f\). Thus, the function \(g(\cdot)\) in Equations (17) and (18) has its approximated formulation as \(f(\alpha)=f(\beta)+\left\langle\
abla f(\beta),\alpha-\beta\right\rangle\), where \(\beta\in R^{n}\) is sufficiently close to \(\alpha\). By using first-order Taylor expansion, \(\
abla f(\beta)\) represents the gradient of \(f(\cdot)\) at \(\beta\). The same principle can also be implemented for the penalty function \(h_{\lambda}(\cdot)\). Expanding \(g(\mathbf{L})\) and \(f_{\lambda}(\mathbf{S})\) using the first-order Taylor series, the Lagrangian form of the optimization problem described above can be established based on Equation (9). \[ \begin{split}\mathcal{L}(\mathbf{L},\mathbf{S},\mathbf{K},\rho)& =\sum\limits_{j}g(\sigma_{j}^{(k)})+\left\langle\
abla g(\sigma_{j}^{(k)}),( \sigma_{j}-\sigma_{j}^{(k)})\right\rangle\\ +&\lambda(\sum\limits_{ij}f_{\lambda}(s_{ij}^{(k)}) +\left\langle\
abla f_{\lambda}(s_{ij}^{(k)}),(s_{ij}-s_{ij}^{(k)})\right\rangle )\\ +&\left\langle\mathbf{M}-\mathbf{L}-\mathbf{S},\mathbf{K}\right\rangle +\frac{\rho^{(k)}}{2}\|\mathbf{M}-\mathbf{L}-\mathbf{S}\|_{F}^{2}\end{split}  \tag{19}\]  
where the dual variable \(K\) is the Lagrange multiplier and \(\rho\) is the penalty coefficient. For the sake of convenience in subsequent calculations and processing, \(\left\langle\mathbf{M}-\mathbf{L}-\mathbf{S},\mathbf{K}\right\rangle\) and \(\frac{\rho^{(k)}}{2}\|\mathbf{M}-\mathbf{L}-\mathbf{S}\|_{F}^{2}\) can be combined. \[