Phase Retrieval from Local Measurements: Improved Robustness via Eigenvector-Based Angular Synchronization

TL;DR

This paper presents an improved phase retrieval method using eigenvector-based angular synchronization, offering enhanced robustness, efficiency, and theoretical guarantees, applicable to local and Fourier-based measurements.

Contribution

It introduces a robust, fast algorithm for phase retrieval with deterministic measurement constructions and improved error guarantees, extending to windowed Fourier measurements.

Findings

Outperforms competing methods in robustness and speed

Provides improved theoretical reconstruction error guarantees

Extends naturally to windowed Fourier measurement scenarios

Abstract

We improve a phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks [27]. The improved algorithm admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees from [27] are improved as a result for the new and more robust reconstruction approach proposed herein. Numerical experiments demonstrate robustness and computational efficiency that outperforms competing approaches on large problems. Finally, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.