Convergence of the randomized Kaczmarz method for phase retrieval

TL;DR

This paper proves that the randomized Kaczmarz method converges exponentially for phase retrieval in real space, with high probability, when using a measurement system of size proportional to the dimension.

Contribution

It provides the first rigorous convergence guarantee for the randomized Kaczmarz method applied to phase retrieval in real space.

Findings

Convergence in the mean square sense is guaranteed with high probability.

The convergence rate is exponential and comparable to the classical linear case.

A measurement system of size proportional to the dimension suffices for convergence.

Abstract

The classical Kaczmarz iteration and its randomized variants are popular tools for fast inversion of linear overdetermined systems. This method extends naturally to the setting of the phase retrieval problem via substituting at each iteration the phase of any measurement of the available approximate solution for the unknown phase of the measurement of the true solution. Despite the simplicity of the method, rigorous convergence guarantees that are available for the classical linear setting have not been established so far for the phase retrieval setting. In this short note, we provide a convergence result for the randomized Kaczmarz method for phase retrieval in $\mathbb{R}^d$. We show that with high probability a random measurement system of size $m \asymp d$ will be admissible for this method in the sense that convergence in the mean square sense is guaranteed with any prescribed probability. The convergence is exponential and comparable to the linear setting.