🤖 AI Summary
This paper addresses the affine phase retrieval problem—reconstructing an unknown signal from the magnitudes of affine measurements. To tackle this nonconvex, nonsmooth inverse problem, we propose a second-order optimization framework integrating Newton’s method and the Gauss–Newton method. Under signal priors (e.g., sparsity or structural constraints) and measurement models (e.g., Gaussian random or coded diffraction patterns), we establish, for the first time, a global quadratic convergence theory: we prove strong convexity of the objective function in a neighborhood of the solution and provide a unified convergence analysis for both second-order methods. The theoretical guarantees are initialization-free and hold in the noiseless setting. Numerical experiments demonstrate that our approach outperforms state-of-the-art first-order algorithms in reconstruction accuracy, convergence speed, and sampling efficiency—achieving exact recovery with measurements nearly attaining the information-theoretic lower bound, thus offering both computational efficiency and robustness.
📝 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.