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This paper solves the problem of block sparse vector recovery using the block ℓ1−αℓq$\ell _1-\alpha \ell _q$‐minimization model. Based on the block restricted isometry property (B‐RIP) condition, exact block sparse vector recovery result is obtained. The theoretical bound for the block ℓ1−αℓq$\ell _1-\alpha \ell _q$‐minimization model are also obtained when measurements are depraved by the noises. [ABSTRACT FROM AUTHOR]

This paper focuses on the compressed sensing ℓ1−ℓ2$\ell _1-\ell _2$‐minimization model and develops new bounds on cumulative coherence μ1(s)$\mu _1(s)$. It is pointed out that if cumulative coherence μ1(s)$\mu _1(s)$ satisfies Equation (2) or (11), then the sparse signal can stably recover in noise model and exactly recover in free noise by ℓ1−ℓ2$\ell _1-\ell _2$‐minimization model. From this paper, it is found that based on some condition of cumulative coherence, the ℓ1−ℓ2$\ell _1-\ell _2$‐minimization model can exactly recover s‐sparse signals in noiseless cases and stably recover s‐sparse signals in the noise cases. [ABSTRACT FROM AUTHOR]

In the field of compressed sensing, ℓ1−2$\ell _{1-2}$‐minimization model can recover the sparse signal well. In dealing with the ℓ1−2$\ell _{1-2}$‐minimization problem, most of the existing literature uses the difference of convex algorithm (DCA) to solve the unrestricted ℓ1−2$\ell _{1-2}$‐minimization model, that is, model (4). Although experiments have proved that the unrestricted ℓ1−2$\ell _{1-2}$‐minimization model can recover the original sparse signal, the theoretical proof has not been established yet. This paper mainly proves theoretically that the unrestricted ℓ1−2$\ell _{1-2}$‐minimization model can recover the sparse signal well, and makes an experimental study on the parameter λ in the unrestricted minimization model. The experimental results show that increasing the size of parameter λ in (4) appropriately can improve the recovery success rate. However, when λ is sufficiently large, increasing λ will not increase the recovery success rate. [ABSTRACT FROM AUTHOR]