# Stable Analysis of Compressive Principal Component Pursuit
## 1 Introduction  
Recently, there has been a rapidly increasing interest in recovering a target matrix that is a superposition of low-rank and sparse components from a small set of linear measurements. In many cases, this problem is shorted for matrix completion [1; 2; 3], which arises in a number of fields, such as medical imaging [4; 5], seismology [6], and computer vision [7; 8] and Kalman filter [9]. Mathematically, there exists a large-scale data matrix \(M=L_{0}+S_{0}\), where \(L_{0}\) is a low-rank matrix, and \(S_{0}\) is a sparse matrix. One of the important problems here is how to extract the intrinsic low-dimensional structure from a small set of linear measurements. In a recent paper [10], [PERSON] et al. proved that most low-rank matrices and the sparse components can be recovered, provided that the rank of the low-rank component is not too large, and that the sparse component is reasonably sparse. It is more important that they proved that these two components can be recovered by solving a simple convex optimization problem. In [11], [PERSON] et al. generalized this problem to decompose a matrix into multiple incoherent components:  
\[\begin{array}{ll}\text{minimize}&\sum\limits_{i}^{\tau}{{{\lambda}_{i}}{{ \left\|{{X}_{i}}\right\|}_{(i)}}}\\ \text{subject to}&\sum\limits_{i}^{\tau}{{{X}_{i}}}=M\,,\end{array} \tag{1}\]  
where \({{\left\|{{X}_{i}}\right\|}_{(i)}}\) are norms that encourage various types of low-complexity structure. The authors also provide a sufficient condition that can promise the existence and uniqueness theorem of compressive principle component pursuit (CPCP). The result in [11] requires that the components are low-complexity structures.  
However, in many applications, the observed measurements are always corrupted by different kinds of noise which may affect every entry of the data matrix. In order to further complete the theorydeveloped in [11], it is necessary to research the stability of CPCP which can guarantee stable and accurate recovery in the presence of entry-wise noise. In this paper, we make a commendable attempt in this respect. We denote \(M\) as the observing matrix which can decompose into multiple incoherent components, and assume that  
\[M=\sum_{i}^{\tau}X_{i,0}+Z_{0}\,,\]  
where \(X_{i,0}\) are corresponding incoherent components and \(Z_{0}\) is an independent and identically distributed (i.i.d.) noise. We assume that \(Z_{0}\) is only limited by \(\|Z_{0}\|_{F}\leq\delta\) for some \(\delta>0\). In order to recover the unknown low-complexity structures, we suggest solving the following relaxed optimization problem.  
\[\begin{array}{ll}\text{minimize}&\sum\limits_{i}^{\tau}\lambda_{i}\|X_{i} \|_{(i)}\\ \text{subject to}&\|\sum\limits_{i}^{\tau}X_{i}-M\|_{F}\leq\delta\end{array} \tag{2}\]  
In this paper, we prove the solution of (2) is stable to small entry-wise noise. The rest of paper is organized as follows. In Section 2, we show some notations and the main result, which will be proven in Sections 3 and 4. In Section 3, we give two important lemmas which are an important parts of our main result. In Section 4, The proof of Theorem 1 will be given. We further provide numerical results in Section 5 and conclude the paper in Section 6.