Perturbation analysis of 𝐿1‒2 method for robust sparse recovery

Abstract

Abstract One of the crucial challenges in applying the theory of compressed sensing in practice is to cope with the perturbation caused by different types of unknown noise, which may arise from the physical implementation or the human mis-modeling. In this paper, we study the robust recovery from a completely perturbed model via a recently popular L 1 - 2 L_{1-2} method. By using the powerful restricted isometry constant of order t ⁢ k tk for t > 1 t>1 , we first obtain a series of perturbation analysis results for this L 1 - 2 L_{1-2} method, which shows that this method is also able to guarantee a robust recovery for any (nearly) sparse signals when both the ideal observations and the exact measurement matrix are perturbed by the unknown noise. Moreover, one of the established recovery conditions under the noise-free settings is also demonstrated to be much better than the state-of-the-art one. Finally, some simulation experiments are further carried out to verify the effectiveness of this L 1 - 2 L_{1-2} method.