Perturbed Amplitude Flow for Phase Retrieval

Abstract

In this paper, we propose a new non-convex algorithm for solving the phase retrieval problem, i.e., the reconstruction of a signal $ \vx\in\H^n $ ($\stackrel{˝}{=}\mathbb{R}$ or $\C$) from phaseless samples $ b_j=\abs{\langle \va_j, \vx\rangle } $, $j=1,\dots ,m$. The proposed algorithm solves a new proposed model, perturbed amplitude-based model, for phase retrieval and is correspondingly named as {\em Perturbed Amplitude Flow} (PAF). We prove that PAF can recover $c\vx$ ($\abs{c} = 1$) under $\mathcal{O}(n)$ 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.

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Cited by in corpus (2)

• Phase Retrieval via Smooth Amplitude Flow
• Performance bound of the intensity-based model for noisy phase retrieval