The global landscape of phase retrieval II: quotient intensity models

2021/12/15 by Jian‐Feng Cai, Cai, Jian-Feng, Meng Huang +5 · 1 citation
Computer Science · 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

paper · pdf · doi:10.48550/arxiv.2112.07997

openalex publication_date 2021/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A fundamental problem in phase retrieval is to reconstruct an unknown signal from a set of magnitude-only measurements. In this work we introduce three novel quotient intensity-based models (QIMs) based a deep modification of the traditional intensity-based models. A remarkable feature of the new loss functions is that the corresponding geometric landscape is benign under the optimal sampling complexity. When the measurements a_i∈ \Rn are Gaussian random vectors and the number of measurements m≥ Cn, the QIMs admit no spurious local minimizers with high probability, i.e., the target solution x is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Such benign geometric landscape allows the gradient descent methods to find the global solution x (up to a global phase) without spectral initialization.

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• Phase retrieval and matrix sensing via benign and overparametrized nonconvex optimization

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