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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 aiRn are Gaussian random vectors and the number of measurements mCn, 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 Email

Meng Huang Email

Dong Li Email

Yang Wang Email

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