Stochastic Amplitude Flow for phase retrieval, its convergence and doppelg\"angers

TL;DR

This paper establishes convergence guarantees for Stochastic Amplitude Flow (SAF), a stochastic gradient descent method for phase retrieval, and applies these results to related algorithms like Kaczmarz and Ptychographic Iterative Engine.

Contribution

It provides the first convergence analysis for SAF and extends these results to related phase retrieval algorithms, bridging stochastic and deterministic methods.

Findings

SAF converges to critical points under certain conditions.

The analysis applies to Kaczmarz and Ptychographic Iterative Engine.

The results enhance understanding of stochastic methods in phase retrieval.

Abstract

In this paper, we focus on Stochastic Amplitude Flow (SAF) for phase retrieval, a stochastic gradient descent for the amplitude-based squared loss. While the convergence to a critical point of (nonstochastic) Amplitude Flow is well-understood, SAF is a much less studied algorithm. We close this gap by deriving the convergence guarantees for SAF based on the contributions for Amplitude Flow and analysis for stochastic gradient descent. These results are then applied to two more algorithms, which can be seen as instances of SAF. The first is an extension of the Kaczmarz method for phase retrieval. The second is Ptychographic Iterative Engine, which is a popular algorithm for ptychography, a special case of phase retrieval with the short-time Fourier transform. Keywords: phase retrieval, Amplitude Flow, stochastic gradient descent, ptychography, Ptychographic Iterative Engine, Kaczmarz method.