The Global Landscape of Phase Retrieval II: Quotient Intensity Models

The Global Landscape of Phase Retrieval II: Quotient Intensity Models

Year:    2022

Author:    Jian-Feng Cai, Meng Huang, Dong Li, Yang Wang

Annals of Applied Mathematics, Vol. 38 (2022), Iss. 1 : pp. 62–114

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 models (QIMs) based on 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\in \mathbb{R}^n$ are Gaussian random vectors and the number of measurements $m\ge Cn$, the QIMs 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. 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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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aam.OA-2021-0010

Annals of Applied Mathematics, Vol. 38 (2022), Iss. 1 : pp. 62–114

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    53

Keywords:    Phase retrieval landscape analysis non-convex optimization.

Author Details

Jian-Feng Cai

Meng Huang

Dong Li

Yang Wang

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