$k$-sparse signal recovery via unrestricted $\ell_{1-2}$-minimization

In the field of compressed sensing, $\ell_{1-2}$-minimization model can recover the sparse signal well. In dealing with the $\ell_{1-2}$-minimization problem, most of the existing literatures use the DCA algorithm to solve the unrestricted $\ell_{1-2}$-minimization model, i.e. model $(\ref{my1})$. Although experiments have proved that the unrestricted $\ell_{1-2}$-minimization model can recover the original sparse signal, the theoretical proof has not been established yet. This paper mainly proves theoretically that the unrestricted $\ell_{1-2}$-minimization model can recover the sparse signal well, and makes an experimental study on the parameter $\lambda$ in the unrestricted minimization model. The experimental results show that increasing the size of parameter $\lambda$ in model $(\ref{my1})$ appropriately can improve the recovery success rate. However, when $\lambda$ is sufficiently large, increasing $\lambda$ will not increase the recovery success rate.