On Gradient Descent Algorithm for Generalized Phase Retrieval Problem

In this paper, we study the generalized phase retrieval problem: to recover a signal $\bm{x}\in\mathbb{C}^n$ from the measurements $y_r=\lvert \langle\bm{a}_r,\bm{x}\rangle\rvert^2$, $r=1,2,\ldots,m$. 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 $\bm{x}$ is real-valued, we prove that the cost function is local convex near the solution $\{\pm\bm{x}\}$. 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.