In this paper, we establish a new compressed sensing model, specifically the $\ell_1-\ell_G$-minimization model. We derive the necessary and sufficient conditions for the recovery of fixed sparse vectors using the $\ell_1-\ell_G$ local minimization model. Building upon this foundation, we further obtain its equivalent condition.

This paper solves the problem of block sparse vector recovery using the block $\ell_1-\alpha\ell_q$- minimization model. Based on the block restricted isometry property (B-RIP) condition, we obtain exact block sparse vector recovery result. We also obtain the theoretical bound for the block $\ell_1-\alpha\ell_q$- minimization model when measurements are depraved by the noises.

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.