Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms

TL;DR

This paper introduces a geometric analysis of polytope-based algorithms for phase retrieval, deriving performance guarantees, revealing phase transitions, and proposing new algorithms that outperform existing methods in various measurement scenarios.

Contribution

It provides a sharp geometric characterization of the polytope used in phase retrieval, leading to new algorithms with improved recovery performance and theoretical guarantees.

Findings

Phase transition identified in sample complexity for PhaseMax.

New nonconvex algorithm PhaseLamp outperforms PhaseMax.

Weighted PhaseLamp improves results under Gaussian and Fourier measurements.

Abstract

We study algorithms for solving quadratic systems of equations based on optimization methods over polytopes. Our work is inspired by a recently proposed convex formulation of the phase retrieval problem, which estimates the unknown signal by solving a simple linear program over a polytope constructed from the measurements. We present a sharp characterization of the high-dimensional geometry of the aforementioned polytope under Gaussian measurements. This characterization allows us to derive asymptotically exact performance guarantees for PhaseMax, which also reveal a phase transition phenomenon with respect to its sample complexity. Moreover, the geometric insights gained from our analysis lead to a new nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm, which we call PhaseLamp. We show that this new algorithm has superior recovery performance over the original PhaseMax method. Finally, as yet another variation on the theme of performing phase retrieval via polytope optimization, we propose a weighted version of PhaseLamp and demonstrate, through numerical simulations, that it outperforms several state-of-the-art algorithms under both generic Gaussian measurements as well as more realistic Fourier-type measurements that arise in phase retrieval applications.