On Gradient Descent Algorithm for Generalized Phase Retrieval Problem

Abstract

In this paper, we study the generalized phase retrieval problem: to recover a signal $\mathbf{x}\in {\mathbb{C}}^n$ from the measurements $y_r=|⟨{\mathbf{a}}_r,\mathbf{x}⟩{|}^2$, $r=1,2,\dots ,m$. The problem can be reformulated as a least-squares minimization problem. Although the cost function is nonconvex, the global convergence of gradient descent algorithm from a random initialization is studied, when $m$ is large enough. We improve the known result of the local convergence from a spectral initialization. When the signal $\mathbf{x}$ is real-valued, we prove that the cost function is local convex near the solution $\{\pm \mathbf{x}\}$. To accelerate the gradient descent, we review and apply several efficient line search methods. We also perform a comparative numerical study of the line search methods and the alternative projection method. Numerical simulations demonstrate the superior ability of LBFGS algorithm than other algorithms.

14 pages, 14 figures

References in corpus (5)

• Relaxed Averaged Alternating Reflections for Diffraction Imaging
• The Complex Gradient Operator and the CR-Calculus
• When Are Nonconvex Problems Not Scary?
• PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
• The local convexity of solving systems of quadratic equations

Cited by in corpus (2)

• Phase Retrieval via Sparse Wirtinger Flow
• Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames