TY - JOUR
T1 - The Global Landscape of Phase Retrieval I:  Perturbed  Amplitude Models
AU - Jian-Feng Cai , 
AU - Meng Huang , 
AU - Dong Li , 
AU - Wang , Yang
JO - Annals of Applied Mathematics
VL - 4
SP - 437
EP - 512
PY - 2021
DA - 2021/12
SN - 37
DO - http://doi.org/10.4208/aam.OA-2021-0009
UR - https://global-sci.org/intro/article_detail/aam/20092.html
KW - Phase retrieval, landscape analysis, non-convex optimization.
AB - <p style="text-align: justify;">A fundamental task in phase retrieval is to recover an unknown signal $x\in\mathbb{R}^n$ from a set of magnitude-only measurements $y_i=|\langle&nbsp;a_i,x\rangle|,$ $ i=1,\cdots,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 \in \mathbb{R}^n$ 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 $ 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&nbsp; state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.</p>