On global convergence of gradient descent algorithms for generalized phase retrieval problem

In this paper, we study the generalized phase retrieval problem: to recover a signal x∈Cn from the measurements yr=|〈ar,x〉|2, r=1,2,…,m. The problem can be reformulated as a least-squares minimization problem. Although the cost function is nonconvex, the global convergence of gradient descent algorithms 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 x is real-valued, we prove that the cost function is local convex near the solution {±x}. To accelerate the gradient descent, we review and apply several efficient line search methods with exact line search stepsize. 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.