The global landscape of phase retrieval I: perturbed amplitude models

TL;DR

This paper introduces two novel perturbed amplitude models for phase retrieval, proving their benign geometric landscape allows simple gradient descent to reliably recover signals without spectral initialization, outperforming existing methods.

Contribution

The paper proposes two new non-convex models for phase retrieval that guarantee no spurious local minima and enable effective gradient descent without spectral initialization.

Findings

No spurious local minima in the proposed models with high probability

Gradient descent with random initialization outperforms spectral methods

Models exhibit a benign geometric landscape facilitating recovery

Abstract

A fundamental task in phase retrieval is to recover an unknown signal $\vx\in \Rn$ from a set of magnitude-only measurements $y_i=\abs{\nj{\va_i,\vx}}, \; i=1,\ldots,m$. In this paper, we propose two novel perturbed amplitude models (PAMs) which have non-convex and quadratic-type loss function. When the measurements $ \va_i \in \Rn$ are Gaussian random vectors and the number of measurements $m\ge Cn$, we rigorously prove that the PAMs admit no spurious local minimizers with high probability, i.e., the target solution $ \vx$ is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Thanks to the well-tamed benign geometric landscape, one can employ the vanilla gradient descent method to locate the global minimizer $\vx$ (up to a global phase) without spectral initialization. We carry out extensive numerical experiments to show that the gradient descent algorithm with random initialization outperforms state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.