# Solving large-scale general phase retrieval problems via a sequence of convex relaxations

## TL;DR

This paper introduces COPR, a convex relaxation-based iterative algorithm for large-scale general phase retrieval, demonstrating improved reliability and speed over existing methods through simulations.

## Contribution

It proposes a novel convex relaxation approach using nuclear norm minimization and an efficient ADMM algorithm for scalable phase retrieval solutions.

## Key findings

- COPR converges linearly or faster in noise-free cases.
- The ADMM algorithm enhances scalability and speed.
- Numerical simulations show superior performance over state-of-the-art methods.

## Abstract

We present a convex relaxation-based algorithm for large-scale general phase retrieval problems. General phase retrieval problems include i.a. the estimation of the phase of the optical field in the pupil plane based on intensity measurements of a point source recorded in the image (focal) plane. The non-convex problem of finding the complex field that generates the correct intensity is reformulated into a rank constraint problem. The nuclear norm is used to obtain the convex relaxation of the phase retrieval problem. A new iterative method, indicated as Convex Optimization-based Phase Retrieval (COPR), is presented, with each iteration consisting of solving a convex problem. In the noise-free case and for a class of phase retrieval problems the solutions of the minimization problems converge linearly or faster towards a correct solution. Since the solutions to nuclear norm minimization problems can be computed using semidefinite programming, and this tends to be an expensive optimization in terms of scalability, we provide a fast ADMM algorithm that exploits the problem structure. The performance of the COPR algorithm is demonstrated in a realistic numerical simulation study, demonstrating its improvements in reliability and speed with respect to state-of-the-art methods.