Robust recovery of nearly-sparse complex signals from phaseless measurements in the presence of noise

Abstract

Reconstruction of sparse complex signals from quadratic magnitude-only measurements $y_j=|〈a_j,x〉{|}^2+z_j(j=1,2,\cdots ,m)$ is of particular importance in many fields. In this paper, the recovery of nearly k-sparse complex signals from phaseless measurements is examined by ${\ell }_p$ nonconvex minimization method. Sufficient condition is established to guarantee robust and stable recovery of nearly k-sparse signals in different types of noise settings. The new results substantially generalize and improve the state-of-the-art results which focused on the recovery of strictly k-sparse signals from phaseless measurements in noiseless setting.