Affine Phase Retrieval via Second-Order Methods

TL;DR

This paper introduces second-order optimization methods for affine phase retrieval, proving their quadratic convergence and demonstrating their efficiency and robustness through extensive numerical experiments.

Contribution

It develops and analyzes Newton and Gauss-Newton methods for affine phase retrieval, establishing their strong convexity and quadratic convergence under certain conditions.

Findings

Newton method with resampling achieves global quadratic convergence in noiseless setting

Gauss-Newton method also exhibits quadratic convergence

Second-order methods recover signals efficiently with fewer measurements

Abstract

In this paper, we study the affine phase retrieval problem, which aims to recover signals from the magnitudes of affine measurements. We develop second-order optimization methods based on Newton and Gauss-Newton iterations and establish that, under specific a priori conditions, the problem exhibits strong convexity. Theoretically, we prove that the Newton method with resampling achieves global quadratic convergence in the noiseless setting for both Gaussian measurements and admissible coded diffraction patterns (CDPs). Furthermore, we demonstrate that the same theoretical framework naturally extends to the Gauss-Newton method, implying its quadratic convergence. To validate our theoretical findings, we conduct extensive numerical experiments. The results confirm the quadratic convergence of second-order methods, while their computational efficiency remains comparable to that of first-order methods. Additionally, our experiments demonstrate that second-order methods achieve exact recovery with relatively few measurements, highlighting their practical feasibility and robustness.