# Perturbed Amplitude Flow for Phase Retrieval

## TL;DR

This paper introduces Perturbed Amplitude Flow (PAF), a simple, efficient non-convex algorithm for phase retrieval that guarantees linear convergence with optimal measurements and is validated through simulations and image experiments.

## Contribution

The paper presents PAF, a novel non-convex phase retrieval algorithm with proven recovery guarantees and linear convergence, requiring no truncation or re-weighting.

## Key findings

- PAF recovers signals with optimal O(n) measurements.
- PAF converges linearly from a designed initial point.
- Validated effectiveness through simulations and natural image experiments.

## 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 $ ($\H=\R$ or $\C$) from phaseless samples $ b_j=\abs{\langle \va_j, \vx\rangle } $, $ j=1,\ldots,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.