The minimal measurement number for low-rank matrix recovery

doi:10.1016/j.acha.2017.01.005

	The minimal measurement number for low-rank matrix recovery
	Xu, Zhiqiang
	2018-03-01
发表期刊	APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
ISSN	1063-5203
卷号	44 期号:2 页码:497-508
摘要	The 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.
关键词	Low-rank matrices Phase retrieval Determinant variety
DOI	10.1016/j.acha.2017.01.005
语种	英语
资助项目	NSFC[11422113] ; NSFC[91630203] ; NSFC[11021101] ; NSFC[11331012] ; National Basic Research Program of China (973 Program)[2015CB856000]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000423781300011
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/29544
专题	计算数学与科学工程计算研究所
通讯作者	Xu, Zhiqiang
作者单位	Chinese Acad Sci, Acad Math & Syst Sci, LSEC, Inst Comp Math, Beijing 100091, Peoples R China
推荐引用方式 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.