Rip Analysis for the Weighted ℓr −ℓ1 Minimization Method

The weighted ℓr − ℓ1 minimization method with 0 < r ≤ 1 largely generalizes the classical ℓr 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 ℓr − ℓ1 minimization problem via the difference of convex functions algorithms (DCA) directly. Numerical experiments show that our DCA based weighted ℓr − ℓ1 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 ℓ1 − ℓ2 minimization method