The Global Landscape of Phase Retrieval I:Perturbed Amplitude Models

A fundamental task in phase retrieval is to recover an unknown signal x∈R^(n) from a set of magnitude-only measurements y_(i)=|〈a_(i),x〉|,i=1,…,m.In this paper,we propose two novel perturbed amplitude models(PAMs)which have a non-convex and quadratic-type loss function.When the measurements a_(i)∈R^(n) are Gaussian random vectors and the number of measurements m≥Cn,we rigorously prove that the PAMs admit no spurious local minimizers with high probability,i.e.,the target solution x is the unique local minimizer(up to a global phase)and the loss function has a negative directional curvature around each saddle point.Thanks to the well-tamed benign geometric landscape,one can employ the vanilla gradient descent method to locate the global minimizer x(up to a global phase)without spectral initialization.We carry out extensive numerical experiments to show that the gradient descent algorithm with random initialization outperforms state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.