Perturbed Amplitude Flow for Phase Retrieval

In this paper, we propose a new non-convex algorithm for solving the phase retrieval problem, i.e., the reconstruction of a signal x ∈ H
ⁿ
(H = R or C) from phaseless samples b
_j
= |〈a
_j
, x〉|, j = 1, . . . , m. The proposed algorithm solves a new proposed model, perturbed amplitude-based model, for phase retrieval, and is correspondingly named as Perturbed Amplitude Flow (PAF). We prove that PAF can recover cx (|c| = 1) under O(n) Gaussian random measurements (optimal order of measurements). Starting with a designed initial point, our PAF algorithm iteratively converges to the true solution at a linear rate for both real, and complex signals. Besides, PAF algorithm needn't any truncation or re-weighted procedure, so it enjoys simplicity for implementation. The effectiveness, and benefit of the proposed method are validated by both the simulation studies, and the experiment of recovering natural images.