Convergence analysis of Wirtinger Flow for Poisson phase retrieval

TL;DR

This paper provides a rigorous convergence analysis of the Wirtinger Flow algorithm for Poisson phase retrieval, demonstrating linear convergence and robustness without complex modifications, and introduces an efficient incremental variant.

Contribution

It offers the first comprehensive convergence proof for WF in Poisson phase retrieval and proposes an incremental version that enhances efficiency and guarantees convergence.

Findings

WF achieves linear convergence in noiseless conditions

WF remains stable under bounded noise

Incremental WF improves computational efficiency and converges reliably

Abstract

This paper presents a rigorous theoretical convergence analysis of the Wirtinger Flow (WF) algorithm for Poisson phase retrieval, a fundamental problem in imaging applications. Unlike prior analyses that rely on truncation or additional adjustments to handle outliers, our framework avoids eliminating measurements or introducing extra computational steps, thereby reducing overall complexity. We prove that WF achieves linear convergence to the true signal under noiseless conditions and remains robust and stable in the presence of bounded noise for Poisson phase retrieval. Additionally, we propose an incremental variant of WF, which significantly improves computational efficiency and guarantees convergence to the true signal with high probability under suitable conditions.