## 2 Smoothed L0 Norm Algorithm  
In [10; 11; 12], there is Algorithm 1 SL0 (smoothed \(L0\) norm), which implements a very fast method for searching for a sparse solution to an under-determined system of linear equations, including for data belonging to a complex space. The SL0 algorithm implements an approximate solution based on the gradient method, with a step-by-step projection of the result onto the space of feasible vectors [10]:  
```
1. Start : \(b\), \(A\), \(\sigma_{0}\), \(\mu\), \(\alpha\)
2. Initialization : \(x_{1}=x_{0}\)
3. Cycle by \(l\) with a set number of repetitions \(L\)
4. \(\sigma_{l}=\alpha\cdot\sigma_{l-1}\)
5. Cycle by \(k\) with a set number of repetitions \(K\): \(\Delta x=x_{k}\cdot\exp(\left|x_{k}\right|^{2}/2\sigma_{l}^{2})\)

where \(K>>L\), \(A\) is the selected matrix of the redundant basis, \(x\) is the solution vector, \(\sigma\), \(\mu\), and \(\alpha\leq 1\) are coefficients determining the speed of convergence, and \(H\) is the Hermitian conjugation symbol. This algorithm was compared in [10] with L1-magic, one of the fastest implementations of the linear programming method (LP), and another fast iterative Re-weighted least squares algorithm (IRLS), using FOCUSS. The simulation results presented in [10] demonstrate the following relative convergence times (for SL0, the convergence time is assumed to be 1), as shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Convergence Times** & **Mean Square Error** \\ \hline \multicolumn{3}{c}{**(MSE)**} \\ \hline SL0 & 1 & \(5.53\times 10^{-5}\) \\ LP (l\({}_{1}\)-magic) & 133 & \(2.31\times 10^{-4}\) \\ FOCUSS & 91 & \(6.45\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relative ratios of convergence times for a signal vector length of 1000 and a number of basis functions of 400. The results of Table 1, as reported in [10], show that SL0 operates at a speed that is two orders of magnitude higher than that of LP or FOCUS, while achieving higher MSE values. The change in the normalized convergence rate for various modifications of SL0 is not significant and varies between 0.5 and 1. ## 3 Modified MFNN Algorithm

Along with the methods mentioned above, the decomposition of data into weighted components can be represented as a so-called \"fuel minimization\" problem or MFNN (minimum fuel neural network), as presented in [20], and its modified version for complex-valued data, proposed by the authors [21]. Cosine functions are used as a dictionary in [20]. It is shown that in the field of real numbers, \(\mathbb{R}\), the neural network described by the system of differential equations gives a solution that is globally stable according to Lyapunov and approaches the exact solution of the problem (1). Despite the interesting results obtained, this approach has not been thoroughly investigated to date. The MFNN algorithm is described in more detail in [20], which provides a theoretical justification for an approach based on solving differential equations. The modified MFNN algorithm provides a solution for complex-valued data using a neural network approach with a modified initialization function, as described in [21]. It was demonstrated that by altering the initialization function for a neural network proposed in the paper, complex-valued data could be successfully represented using complex-valued basis elements. The Algorithm 2 modified MFNN implements a solution for complex-valued data based on the neural network method:
```
1. Start: \(x,y,A,b\in\mathbb{C}^{N}\)
2. Initialization : \(x=0,y=0\)
3. Numerical solution of a system of differential equations using a neural network: \(\left\{ \begin{aligned} &\frac{dx}{dt}=-A^{T}[Ax-y-b]-P_{\Omega} \big{(}x+A^{T}y\big{)},\\ &\frac{dy}{dt}=-A\big{[}x+A^{T}y-P_{\Omega}\big{(}x+A^{T}y\big{)} \big{]}+b,\end{aligned} \right.\)  where \(P_{\Omega}(z)=[P_{\Omega}(z_{1}),P_{\Omega}(z_{2}),\dots,P_{\Omega}(z_{n})] ^{T}\), \(P_{\Omega}(z_{i})=\left\{