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The minimal measurement number for low-rank matrix recovery
Xu, Zhiqiang
AbstractThe paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that 4nr - 4r(2) generic measurements are both necessary and sufficient for the recovery of rank-r matrices in C-nxn. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound 4nr - 4r(2) is tight provided n = 2(k) + r, k is an element of Z(+) Motivated by Vinzant's work [19], we construct 11 matrices in R-4 X 4 by computer random search and prove they define injective measurements on rank-1 matrices in R-4x4. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2n - 1 orthogonal projections are possible for the recovery of x is an element of R-n from the norm of them, which gives a negative answer for a question raised in [1]. (C) 2017 Elsevier Inc. All rights reserved.
KeywordLow-rank matrices Phase retrieval Determinant variety
Funding ProjectNSFC[11422113] ; NSFC[91630203] ; NSFC[11021101] ; NSFC[11331012] ; National Basic Research Program of China (973 Program)[2015CB856000]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000423781300011
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Document Type期刊论文
Corresponding AuthorXu, Zhiqiang
AffiliationChinese Acad Sci, Acad Math & Syst Sci, LSEC, Inst Comp Math, Beijing 100091, Peoples R China
Recommended Citation
GB/T 7714
Xu, Zhiqiang. The minimal measurement number for low-rank matrix recovery[J]. APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS,2018,44(2):497-508.
APA Xu, Zhiqiang.(2018).The minimal measurement number for low-rank matrix recovery.APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS,44(2),497-508.
MLA Xu, Zhiqiang."The minimal measurement number for low-rank matrix recovery".APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 44.2(2018):497-508.
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