On phaseless compressed sensing with partially known support

Ying Zhang; Ling Ma; Zheng-Hai Huang

doi:10.3934/jimo.2019014

Article Contents

2020, Volume 16, Issue 3: 1519-1526. Doi: 10.3934/jimo.2019014

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On phaseless compressed sensing with partially known support

School of Mathematics, Tianjin University, Tianjin 300072, China

^* Corresponding author: ZHENG-HAI HUANG
^* Corresponding author: ZHENG-HAI HUANG

Received: April 30, 2018

Revised: September 30, 2018

Early access: March 2019

Published: March 2020

This work was supported by the China Scholarship Council (Grant No. 201706255092) and the National Natural Science Foundation of China (Grant Nos. 11201332, 11431002 and 11871051)

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Abstract

We establish a theoretical framework for the problem of phaseless compressed sensing with partially known signal support, which aims at generalizing the Null Space Property and the Strong Restricted Isometry Property from phase retrieval to partially sparse phase retrieval. We first introduce the concepts of the Partial Null Space Property (P-NSP) and the Partial Strong Restricted Isometry Property (P-SRIP); and then show that both the P-NSP and the P-SRIP are exact recovery conditions for the problem of partially sparse phase retrieval. We also prove that a random Gaussian matrix $ A\in \mathbb{R}^{m\times n} $ satisfies the P-SRIP with high probability when $ m = O(t(k-r)\log(\frac{n-r}{t(k-r)})). $

Keywords:

Phase retrieval,
compressed sensing,
phaseless compressed sensing,
partial null space property,
partial strong restricted isometry property.

Mathematics Subject Classification: Primary: 90C90; Secondary: 94A12.

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