The Improved Bounds of Restricted Isometry Constant for Recovery via $\ell _{p}$ -Minimization

Nonconvex ℓ_p-minimization with p ∈ (0,1) has been studied recently in the context of compressed sensing. In this paper, we prove that as long as the sensing matrix A ∈ R^m×n satisfies restricted isometry property with δ_2k ∈ (0,1), every k-sparse signal x ∈ Rⁿ can be recovered exactly from linear measurement y=Ax via solving some ℓ_p-minimization problem. In fact, it is shown that p <; min {1,1.0873(1-δ_2k)} suffices for the exact k-sparse recovery of ℓ_p-minimization, which improves the existing results greatly.