Solving a perturbed amplitude-based model for phase retrieval

In this paper, we propose a new algorithm for solving phase retrieval problem, i.e., the reconstruction of a signal $ vxinH^n $ ($H=R$ or $C$) from phaseless samples $ b_j=abs{langle va_j, vxrangle } $, $ j=1,ldots,m $. The proposed algorithm solves a perturbed amplitude-based model for phase retrieval and is correspondingly named as {em Perturbed Amplitude Flow } (PAF). We prove that PAF can recover $cvx$ ($abs{c} = 1$) under $O(n)$ Gaussian random measurements (optimal order of measurements). Unlike several existing $O(n)$ algorithms that can be theoretically proven to recover only real signals, our algorithm works for both real and complex signals. Starting with a designed initial point, our PAF algorithm iteratively converges to the true solution at a linear rate. Besides, PAF algorithm enjoys both low computational complexity and the simplicity for implementation. The effectiveness and benefit of the proposed method can be validated by both the simulation studies and the experiment of recovering natural image.