Sparse recovery in probability via l q -minimization with Weibull random matrices for 0 < q ≤ 1

Abstract

Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the l2,q robust null space property that can weaken the D-RIP, and show that Weibull random matrices satisfy the l2,q robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the l q quotient property with high probability. Finally, with the combination of the above mentioned properties, we give two important approximation characteristics of the solutions to the l q -minimization with Weibull random matrices, one is on the stability estimate when the measurement noise e ∈ ℝ n needs a priori ||e||2 ≤ є, the other is on the robustness estimate without needing to estimate the bound of ||e||2. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.