The global landscape of phase retrieval I: perturbed amplitude models

A fundamental task in phase retrieval is to recover an unknown signal $\vx\in \Rn$ from a set of magnitude-only measurements $y_i=\abs{\nj{\va_i,\vx}}, \; i=1,\ldots,m$. In this paper, we propose two novel perturbed amplitude models (PAMs) which have non-convex and quadratic-type loss function. When the measurements $ \va_i \in \Rn$ are Gaussian random vectors and the number of measurements $m\ge Cn$, we rigorously prove that the PAMs admit no spurious local minimizers with high probability, i.e., the target solution $ \vx$ is the unique global 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 $\vx$ (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.