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, the restricted block $\ell_1-\ell_2$ minimization model can recover the block sparse vector well. When solving the restricted block $\ell_1-\ell_2$ minimization model, it is often transformed into a unrestricted $\ell_1-\ell_2$ minimization model, and then the convex algorithm is used to solve the new model. Experiments have shown that this method is effective, but the theoretical results of the unrestricted $\ell_1-\ell_2$ minimization model being able to recover block sparse vectors have not yet been established. The main task of this paper is to establish sufficient conditions for the unrestricted $\ell_1-\ell_2$ minimization model to recover block sparse vectors based on the RIP condition, and to demonstrate the influence of parameter $\lambda$ in the unrestricted $\ell_1-\ell_2$ minimization model on the recovery of block sparse vectors through experimental methods.\\