Riemannian optimization for phase retrieval from masked Fourier measurements

In this paper, we consider the noisy phase retrieval problem under the measurements of Fourier transforms with complex random masks. Here two kinds of Riemannian optimization algorithms, namely, Riemannian gradient descent algorithm (RGrad) and Riemannian conjugate gradient descent algorithm (RCG), are presented to solve such problem from these special but widely used measurements in practical applications. Since the masked Fourier measurements are less random, we establish stable guarantees for signals by the truncated variants of RGrad and RCG, respectively. First of all, a good initialization is constructed by means of a truncated spectral method. Then we prove that a signal x∈ℂn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {x}\in \mathbb {C}^{n}$\end{document} can be recovered robustly to bounded noise through these two algorithms, provided that L=O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L= O(\log n)$\end{document} complex random masks are performed in the measurement process. This implies that the sample complexity is optimal up to a log factor, namely, O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(n\log n)$\end{document}. Particularly, each sequence generated by the truncated RGrad and RCG provably converges to the true solution at a geometric rate in the noiseless case. Finally, several empirical experiments are provided to show the effectiveness and stability of such two kinds of algorithms compared with Wirtinger Flow(WF) algorithm, for which provable guarantee has been set up for signals under masked Fourier measurements, provided that L=O(log4n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L= O(\log ^{4}n)$\end{document}.