## 1 Introduction
\(D=\left(\phi_{\gamma}\right)\) formed dictionary, where \(\phi_{\gamma}\in\mathbb{C}^{M}\) is the component of the basis with indexes \(\gamma\in\Gamma\) and \(\Gamma\subset L^{2}(\mathbb{R})\)  
The input data \(S\) can be represented as:  
\[b=\sum_{i=1}^{N}\alpha_{\gamma_{i}}\phi_{\gamma_{i}}+r^{(N+1)}, \tag{5}\]  
where \(r^{(N+1)}\in\mathbb{C}^{M}\)--the residual vector or error vector--is the result of the approximation of data \(b\) by the basis functions \(\phi_{\gamma_{1}},\phi_{\gamma_{2}},\ldots,\phi_{\gamma_{N}}\) from the dictionary \(D\), and \(\alpha_{\gamma_{i}}\)- \(i\)- is the weight coefficient. The solution to the optimization problem can be obtained by minimizing the norm of the loss function:  
\[J=\left\lVert r^{(N+1)}\right\rVert_{p}=\left\lVert b-Ax\right\rVert_{p}, \tag{6}\]  
where \(\left\lVert\ \right\rVert_{p}\) is the p-norm; in general, \(p=2\) (norm \(l_{2}\)), and \(p=1\) (norm \(l_{1}\)). There are two approaches to solving this problem for \(p=1\). The first approach is described as an MP algorithm. This approach operates with dictionaries obtained from wavelet dictionaries, orthogonal dictionaries, and cosine functions. The second approach is called BP (basis pursuit) and was proposed by [PERSON] and [PERSON] [16]. It operates with a simple dictionary of basis functions and allows the finding of an estimate of \(x\) using the BCSV (Box-Cox stochastic volatility) method by minimizing the \(l_{1}\) norm of \(x\). An implementation option for the Algorithm 1 BP--IRLS (iteratively reweighted least squares) [20] is given below:```
Start: Basis matrix \(A\in\mathbb{R}^{m\times N}\), data vector \(y\in\mathbb{R}^{m}\), initial value of the scales \(\omega_{0}\in\mathbb{R}^{N}\) (by default : \(\omega_{0}=\) (1, 1,..., 1)). \(\varepsilon_{0}=\infty\) for\(k=0\), 1, 2,...do \[x^{k+1}:=\underset{z\in\mathbb{R}^{N}}{\arg\min}z,\text{ diag }(\omega_{k})z\text{ subject to }Az=y,\] \[\varepsilon_{k+1}:=\min\left(\varepsilon_{k},\,\frac{\sigma_{s} \left(\lambda^{k+1}\right)_{\lambda}}{N}\right),\] \[\left(\omega_{k+1}\right)_{i}:=\frac{1}{\max\left\{\left|x_{i}^{k+1} \right|\left|x_{k+1}\right\}}\text{ for each }i\in[N]\] endfor return Sequence \(\left(x^{k}\right)_{k\geq 1}\)
```

**Algorithm 1****BP**

However, the algorithm is still not fast enough, and it does not allow the use of complex data. In [21; 22; 23], Algorithm 2 SL0 (smoothed \(l_{0}\) norm) is presented, which implements a fairly fast method of finding a sparse solution to an uncertain system of linear equations, including for complex data. The SL0 algorithm implements an approximate solution based on the gradient method, with step-by-step projection of the result onto the space of feasible vectors:

```
1. Start: \(b\), \(A\), \(\left[\sigma_{1},\,\ldots,\,\sigma_{K}\right]\), \(\mu\)  Initialization: \(x_{1}=x_{0}\)  Cycle in \(k\) with a given number of repetitions: \[\Delta x=x_{k}\exp\left(\left|x_{k}\right|^{2}/2\sigma_{k}^{2}\right)\] \[x_{k}=\left(x_{k-1}-\mu\cdot\Delta x\right)-A^{H}\big{(}AA^{H}\big{)}^{-1}(Ax_{k -1}-b)\]  
where \(b\) is the initial coefficient of the scales, \(A\) is the selected excess basis matrix, \(\sigma\) and \(\mu\) are coefficients determining the speed of convergence, and \(x\) is the solution vector. However, despite the high speed of convergence, this algorithm is sensitive to the choice of initial values of the coefficients and does not guarantee convergence to the optimal solution. Summarizing the above, we can highlight three characteristic points regarding this algorithm:  
1. Minimizing the loss function leads to minimizing the number of non-zero components in vector \(x\). 2. Based on the perturbation theory, the \(l_{1}\) norm is more robust compared to the \(l_{2}\) norm, which affects the quality of processing of non-stationary signals that contain sections of transient processes. 3. From a computational point of view, the problem belongs to the LP problem (linear programming, the class of optimization problems).