2021/12/15 by Jian‐Feng Cai, Cai, Jian-Feng, Meng Huang +5 · 3 citations
Computer Science · Engineering · Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Numerical Analysis (math.NA) #Optical measurement and interference techniques #Robotics and Sensor-Based Localization
paper · pdf · doi:10.48550/arxiv.2112.07993
openalex publication_date 2021/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A fundamental task in phase retrieval is to recover an unknown signal \vx∈ \Rn from a set of magnitude-only measurements yi=\abs\nj\vai,\vx, i=1,…,m. In this paper, we propose two novel perturbed amplitude models (PAMs) which have non-convex and quadratic-type loss function. When the measurements \vai ∈ \Rn 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 \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.