RIP analysis for the weighted ℓr-ℓ1 minimization method.

• The restricted isometry property (RIP) and high-order RIP analysis results for the weighted ℓ r − ℓ 1 minimization method are presented. • Through a novel decomposition of the objective function into a difference of two convex functions, the weighted ℓ r − ℓ 1 minimization problem is solved via the difference of convex functions algorithms (DCA) directly. • Numerical experiments show that the DCA based weighted ℓ r − ℓ 1 minimization method gives satisfactory results in sparse recovery no matter whether the measurement matrix is coherent or not. • For highly coherent measurements, the proposed method even outperforms the state-of-art ℓ 1 − ℓ 2 minimization method. The weighted ℓ r − ℓ 1 minimization method with 0 < r ≤ 1 largely generalizes the classical ℓ r minimization method and achieves very good performance in compressive sensing. However, its restricted isometry property (RIP) and high-order RIP analysis results remain unknown. In this paper, we fill in this gap by adopting newly developed analysis tools. Moreover, through a novel decomposition of the objective function into a difference of two convex functions, we propose to solve the weighted ℓ r − ℓ 1 minimization problem via the difference of convex functions algorithms (DCA) directly. Numerical experiments show that our DCA based weighted ℓ r − ℓ 1 minimization method gives satisfactory results in sparse recovery no matter whether the measurement matrix is coherent or not. For highly coherent measurements, our proposed method even outperforms the state-of-art ℓ 1 − ℓ 2 minimization method. [ABSTRACT FROM AUTHOR]