On sparse approximations of solutions to linear systems with orthogonal matrices

This article discusses a model for obtaining a sparse representation of a signal vector in $\mathbb{R}^k$, based on a system of linear equations with an orthogonal matrix. Such a representation minimizes a target function that combines the deviation from the exact solution and a chosen functional $J$. The functionals chosen are the Euclidean norm, the norm $|\cdot|_1$, and the quasi-norm $|\cdot|_0$. The Euclidean norm only allows for the exact solution, while the other two allow for a balance between the residual and the parameter $\lambda$ in the functional, resulting in sparser solutions. Graphs are plotted showing the dependence between the coordinates of the optimal vector and the parameter $\lambda$, and examples are provided.