## 2 Background  
### Sparse Representation  
For a signal, if the coefficient vector decomposed by a suitable dictionary meets the following properties: if all the other elements are zero except that the value of a few elements is non-zero, the signal can be sparse represented, and the coefficient vector obtained by decomposing is called the sparse representation vector of the original signal. Sparse representation is the most representative method in linear representation. It has also been proved to be a very powerful solution. It shows great potential in signal and image denoising and has been widely used in the fields of machine learning [23] and computer vision [24]. Assuming a given measurement signal \(\mathbf{y}\in\mathbb{R}^{M}\), its sparse representation model [2] is defined as:  
\[\mathbf{y}=\mathbf{D}\mathbf{x}+\mathbf{\varepsilon}=\sum_{k=1}^{K}\mathbf{d}_{k}x_{k}+\mathbf{\varepsilon} \tag{1}\]  
where \(\mathbf{D}\in\mathbb{R}^{M\times K}\) is the dictionary matrix, \(M\) is the number of rows of dictionary \(\mathbf{D}\), \(K\) is the number of columns of dictionary \(\mathbf{D}\); \(\mathbf{d}_{k}\) is a column vector in a dictionary that can be called atom or basis function. \(\mathbf{x}=[x_{1},x_{2},\ldots x_{k}]^{\mathrm{T}}\) is the sparse representation vector of input signal \(\mathbf{y}\), which can be regarded as the weight of base function \(\mathbf{d}_{k}\), \(\mathbf{\varepsilon}\) is noise. In the sparse representation model, the number of columns in dictionary \(\mathbf{D}\) is usually much greater than the number of rows (\(K>>M\)), that is, the dictionary is over-complete or redundant, which leads to the fact that \(\mathbf{x}\) can have infinite solutions. Adding sparse constraints to the model can solve the degradation problem caused by over-completeness. Therefore, the objective function of sparse representation can be expressed as  
\[\operatorname*{argmin}_{x}\tfrac{1}{2}\|\mathbf{y}-\mathbf{D}\mathbf{x}\|_{2}+\lambda\| \mathbf{x}\|_{0} \tag{2}\]  
where \(\|\cdot\|_{0}\) is the pseudo-norm of \(l_{0}\) and the return value is the number of non-zero values of the vector. It is generally considered that \(\|\mathbf{y}-\mathbf{D}\mathbf{x}\|_{2}\) is the data fidelity term, which ensures that the signal reconstructed by the product of the dictionary and coefficient vector is as consistent as possible with the original signal. \(\lambda\|\mathbf{x}\|_{0}\) is a regular term, and its design is generally designed with prior knowledge to constrain parameters in the model. In the sparse representation model, it is a sparse constraint on the decomposed coefficient vector. The methods of solving Equation (2) are mainly divided into two categories: one is a greedy algorithm, such as match pursuit (MP), orthogonal match pursuit (OMP), and piecewise orthogonal match pursuit (StOMP). Since the \(l_{0}\) norm is non-convex, the solution strategy of this kind of algorithm is mainly to use an exhaustive search to solve the sparse representation problem, which is almost an NP-hard problem. With the increase of dimension, the complexity of the solution increases rapidly, resulting in the need to spend a lot of computational cost in the actual situation. Another strategy is to smooth the \(l_{0}\) norm into an \(l_{1}\) norm convex optimization problem. [PERSON] and [PERSON] proved that under RIP conditions, the solution of Equation (2) in the \(l_{0}\) norm is consistent with that in the \(l_{1}\) norm, and the problem is easily solved by a gradient correlation algorithm. \(l_{1}\) norm regularization problem is also called the Lasso regression problem and its objective function is:  
\[\operatorname*{argmin}_{x}\tfrac{1}{2}\|\mathbf{y}-\mathbf{D}\mathbf{x}\|_{2}+\lambda\| \mathbf{x}\|_{1} \tag{3}\]  
where \(\|\mathbf{x}\|_{1}=\sum_{n}|x_{n}|\) is the \(l_{1}\) norm. Typical methods for solving Equation (3) include an iterative shrinkage-thresholding algorithm (ISTA), gradient projection for sparse reconstruction (GPSR), and the interpolation point method, etc.