Sparse Signal Recovery by ${\ell _q}$ Minimization Under Restricted Isometry Property

In the context of compressed sensing, the nonconvex l_q minimization with 0 <; q <; 1 has been studied in recent years. In this letter, by generalizing the sharp bound for l₁ minimization of Cai and Zhang, we show that the condition σ(s^q+1)k <; 1/√(s^q-2+1) in terms of restricted isometry constant (RIC) can guarantee the exact recovery of k-sparse signals in the noiseless case and the stable recovery of approximately k-sparse signals in the noisy case by l_q minimization. This result is more general than the sharp bound for l₁ minimization when the order of RIC is greater than 2k and illustrates the fact that a better approximation to l₀ minimization is provided by l_q minimization than that provided by l₁ minimization.