The global landscape of phase retrieval II: quotient intensity models

TL;DR

This paper introduces three novel quotient intensity-based models for phase retrieval that have a benign geometric landscape, enabling efficient recovery of signals from Gaussian measurements without spectral initialization.

Contribution

The paper proposes three new quotient intensity models with favorable geometric properties, improving phase retrieval by eliminating spurious local minima under Gaussian measurements.

Findings

No spurious local minimizers with high probability when m ≥ Cn

Gradient descent can recover the true signal without spectral initialization

Loss functions have negative curvature around saddle points

Abstract

A fundamental problem in phase retrieval is to reconstruct an unknown signal from a set of magnitude-only measurements. In this work we introduce three novel quotient intensity-based models (QIMs) based a deep modification of the traditional intensity-based models. A remarkable feature of the new loss functions is that the corresponding geometric landscape is benign under the optimal sampling complexity. When the measurements $ a_i\in \Rn$ are Gaussian random vectors and the number of measurements $m\ge Cn$, the QIMs admit no spurious local minimizers with high probability, i.e., the target solution $ x$ is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Such benign geometric landscape allows the gradient descent methods to find the global solution $x$ (up to a global phase) without spectral initialization.