## 4 The Non-Convex Approximate RPCA Algorithm
### Derivation of the Overall Formula for NCA-RPCA Algorithm
\begin{split}&\left\langle\mathbf{M}-\mathbf{L}-\mathbf{S},\mathbf{K}\right\rangle +\frac{\rho^{(k)}}{2}\|\mathbf{M}-\mathbf{L}-\mathbf{S}\|_{F}^{2}\\ &=\left\langle\mathbf{M}-\mathbf{L}-\mathbf{S},\mathbf{K}\right\rangle+\frac{ \rho^{(k)}}{2}\|\mathbf{M}-\mathbf{L}-\mathbf{S}\|_{F}^{2}+\frac{1}{2\rho^{(k)}}\|\mathbf{K}\| _{F}^{2}-\frac{1}{2\rho^{(k)}}\|\mathbf{K}\|_{F}^{2}\\ &=\frac{\rho^{(k)}}{2}\left\|\mathbf{M}-\mathbf{L}-\mathbf{S}+\frac{\mathbf{K}}{ \rho^{(k)}}\right\|_{F}^{2}-\frac{1}{2\rho^{(k)}}\|\mathbf{K}\|_{F}^{2}\\ &\leq\frac{\rho^{(k)}}{2}\left\|\mathbf{M}-\mathbf{L}-\mathbf{S}+\frac{\mathbf{K}} {\rho^{(k)}}\right\|_{F}^{2}\end{split}  \tag{20}\]  
Here, \(\frac{1}{2\rho^{(k)}}\|\mathbf{K}\|_{F}^{2}\) can be treated as a constant when solving for \(L\) and \(S\), thus Equation (20) can be scaled. In other words, \(\left\langle\mathbf{M}-\mathbf{L}-\mathbf{S},\mathbf{K}\right\rangle+\frac{\rho^{(k)}}{2}\| \mathbf{M}-\mathbf{L}-\mathbf{S}\|_{F}^{2}\) can be equivalently represented as \(\frac{\rho^{(k)}}{2}\left\|\mathbf{M}-\mathbf{L}-\mathbf{S}+\frac{\mathbf{K}}{\rho^{(k)}}\right\| _{F}^{2}\). Continuing, the IALM is employed to solve the equation, mainly involving three steps: in the first step, \(L\) and \(K\) are fixed, and \(S\) is updated to find its optimal value. Since \(\sum\limits_{j}g(\sigma_{j}^{(k)})+\left\langle\
abla g(\sigma_{j}^{(k)}),( \sigma_{j}-\sigma_{j}^{(k)})\right\rangle\) is independent of \(S\), it can be omitted during the process of solving for \(S\). \(\sum\limits_{ij}f_{\lambda}(s_{ij}^{(k)})\) is a constant, and it can also be omitted and scaled while solving for \(S\), as follows:  
\[ \begin{split}\mathbf{S}^{(k+1)}&=\operatorname*{argmin }_{\mathbf{S}}\mathcal{L}(\mathbf{L}^{(k)},\mathbf{S},\mathbf{K}^{(k)},\rho^{(k)})\\ &=\operatorname*{argmin}_{\mathbf{S}}\lambda(\sum\limits_{ij}f_{\lambda }(s_{ij}^{(k)})+\left\langle\
abla f_{\lambda}(s_{ij}^{(k)}),(s_{ij}-s_{ij}^ {(k)})\right\rangle)\\ &\qquad\qquad+\frac{\rho^{(k)}}{2}\left\|(\mathbf{M}-\mathbf{L}^{(k)}+ \frac{\mathbf{K}^{(k)}}{\rho^{(k)}})-\mathbf{S}\right\|_{F}^{2}\end{split}  \tag{21}\]  
Because in Equation (32), \(\sum\limits_{ij}f_{\lambda}(s_{ij}^{(k)})+\left\langle\
abla f_{\lambda}(s_{ ij}^{(k)}),(s_{ij}-s_{ij}^{(k)})\right\rangle\) represents the first-order [PERSON] expansion of \(f_{\lambda}(\mathbf{S})\) and \(\sum\limits_{ij}f_{\lambda}(s_{ij}^{(k)})\) is a constant; therefore, during the process of solving for \(S\), \(\sum\limits_{ij}f_{\lambda}(s_{ij}^{(k)})\) can be omitted, meaning that \(\sum\limits_{ij}f_{\lambda}(s_{ij}^{(k)})+\left\langle\
abla f_{\lambda}(s_{ ij}^{(k)}),(s_{ij}-s_{ij}^{(k)})\right\rangle\) is equivalent to \(\left\langle\bigtriangledown f_{\lambda}(s_{ij}^{(k)}),(s_{ij}-s_{ij}^{(k)})\right\rangle\). The above equations are derived and simplified. \[ \begin{array}{ll}\mathbf{S}^{(k+1)}&=\underset{S}{\text{argmin}}\ \mathcal{L}(\mathbf{L}^{(k)},\mathbf{S},\mathbf{K}^{(k)}, \rho^{(k)})\\ &=\underset{S}{\text{argmin}}\ \lambda\Big{\langle}\bigtriangledown f_{\lambda}(s ^{(k)}_{ij}),(s_{ij}-s^{(k)}_{ij})\big{\rangle}+\frac{\rho^{(k)}}{2}\Big{\|} (\mathbf{M}-\mathbf{L}^{(k)}+\frac{\mathbf{K}^{(k)}}{\rho^{(k)}})-\mathbf{S}\Big{\|}_{F}^{2} \end{array}  \tag{22}\]  
In the second step, with \(K\) fixed along with the updated \(S\), the optimization of \(L\) is performed to obtain the optimal value of \(L\). Similarly, it can be observed that \(\lambda(\underset{ij}{\sum}f_{\lambda}(s^{(k)}_{ij})+\Big{\langle}\bigtriangledown f _{\lambda}(s^{(k)}_{ij}),(s_{ij}-s^{(k)}_{ij})\big{\rangle})\) is independent of \(L\) during the process of solving for \(L\), and thus can be omitted. \[