Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study

TL;DR

This paper explores the adaptation of Kaczmarz methods to solve quadratic equations in phase retrieval, demonstrating computational efficiency and empirical success over existing algorithms.

Contribution

It introduces a novel extension of Kaczmarz methods for phase retrieval, incorporating phase selection heuristics and providing preliminary convergence analysis.

Findings

Kaczmarz methods outperform state-of-the-art algorithms in measurement efficiency.

Kaczmarz methods are computationally faster per iteration.

Empirical results show successful phase retrieval with fewer measurements.

Abstract

We study the Kaczmarz methods for solving systems of quadratic equations, i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz methods for solving systems of linear equations by integrating a phase selection heuristic in each iteration and overall have the same per iteration computational complexity. Extensive empirical performance comparisons establish the computational advantages of the Kaczmarz methods over other state-of-the-art phase retrieval algorithms both in terms of the number of measurements needed for successful recovery and in terms of computation time. Preliminary convergence analysis is presented for the randomized Kaczmarz methods.