## 4 The Non-Convex Approximate RPCA Algorithm  
### The Non-Convex Approximation of Rank  
Although RPCA exhibits high robustness, it also has limitations. Firstly, the assumption of a convex nature for the low-rank matrix in RPCA does not hold true in certain cases, which affects the accuracy of the algorithm. Secondly, RPCA involves solving convex optimization problems to determine the low-rank and sparse matrices, and the slow convergence of convex optimization affects the efficiency of RPCA when dealing with large-scale problems. To address these issues, an improvement has been made by introducing non-convex functions to approximate the matrix rank and \(\ell_{0}\)-norm. This enhancement modifies the low-rank and sparse matrices in RPCA and optimizes the solution of convex optimization problems to enhance computational efficiency.  
The optimization model of RPCA is relaxed to  
\[\begin{split}&\min_{\mathbf{L}\mathbf{S}}\left\|\mathbf{L}\right\|_{*}+ \lambda\left\|\mathbf{S}\right\|_{1}\\ & s.t.\mathbf{M}=\mathbf{L}+\mathbf{S}\end{split} \tag{16}\]  
where \(\left\|\mathbf{L}\right\|_{*}=\sum_{i=1}^{r}\sigma_{i}(\mathbf{M})\), \(\sigma_{i}(\mathbf{M})\) represents the \(i\)th singular value of the matrix \(M\).  
Although Equation (16) is a convex problem and easily optimized, in practical scenarios, due to possible data corruption, obtaining the global optimal solution of Equation (16) may lead to significant errors. Furthermore, since the nuclear norm is essentially equivalent to the \(\ell_{1}\)-norm of the vector of matrix singular values, the \(\ell_{1}\)-norm exhibits a shrinkage effect that results in biased estimates. This implies that the nuclear norm excessively penalizes large singular values, causing solutions that deviate significantly from the true solution.  
Therefore, a new non-convex approximation function is employed to approximate the rank of \(L\), which provides a closer approximation than the nuclear norm. The mathematical form of the non-convex function is defined as follows:  
\[rank(\mathbf{L})\approx g(\mathbf{L})=\left\|\mathbf{L}\right\|_{\gamma}=\sum_{i=1}^{m} \frac{\sigma_{i}(\mathbf{L})}{\sqrt{\sigma_{i}(\mathbf{L})^{2}+\gamma}} \tag{17}\]  
where \(\gamma>0\) represents the model parameter, \(\sigma_{i}(\mathbf{L})\) represents the ith singular value of matrix \(\mathbf{L}\).  
The non-convex rank-approximation function y possesses the following characteristics:  
1. \(\lim_{\gamma\to 0}\left\|\mathbf{L}\right\|_{\gamma}=rank(\mathbf{L})\);
2. \(\sigma_{i}(\mathbf{L})=0\), \(g(\sigma_{i}(\mathbf{L}))=0\);
3. \(\left\|\mathbf{L}\right\|_{\gamma}\) is unitary invariant, meaning that for any orthogonal matrix \(\mathbf{U}\in R^{m\times m}\) and \(\mathbf{V}\in R^{m\times n}\), it holds that \(\left\|\mathbf{L}\right\|_{\gamma}=\left\|\mathbf{U}\mathbf{V}\right\|_{\gamma}\);
4. For any matrix \(\mathbf{L}\in R^{m\times n}\), it holds that \(\left\|\mathbf{L}\right\|_{\gamma}\geq 0\), if and only if \(\mathbf{L}=0\), \(\left\|\mathbf{L}\right\|_{\gamma}=0\).  
Figure 4 illustrates the case of using the non-convex function \(\gamma\) and nuclear norm to approximate the rank function, with the value of \(\sigma_{i}\) increasing gradually. It can be observed that as the singular value \(\sigma_{i}(\mathbf{L})\) increases, the nuclear norm deviates significantly from 1, while the non-convex function steadily approaches one more stably. A smaller parameter value \(\gamma\) leads to a better approximation of the rank function.