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Abstract In this paper, we study a nonconvex, nonsmooth, and non-Lipschitz weighted l p − l q $l_{p}-l_{q}$ ( 0 < p ≤ 1 $0< p\leq 1$ , 1 < q ≤ 2 $1< q\leq 2$ ; 0 ≤ α ≤ 1 $0\leq \alpha \leq 1$ ) norm as a nonconvex metric to recover block-sparse signals and rank-minimization problems. Using block-RIP and matrix-RIP conditions, we obtain exact recovery results for block-sparse signals and rank minimization. We also obtain the theoretical bound for the block-sparse signals and rank minimization when measurements are degraded by noise.