Phase retrieval from the norms of affine transformations

doi:10.1016/j.aam.2021.102213

CSpace

	Phase retrieval from the norms of affine transformations
	Huang, Meng 1; Xu, Zhiqiang 1,2
	2021-09-01
发表期刊	ADVANCES IN APPLIED MATHEMATICS
ISSN	0196-8858
卷号	130 页码:20
摘要	In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals x is an element of F-d from the magnitude of the affine transformations y(j) = parallel to M-j* x + b(j) parallel to + 1 )(2)(2) j = 1, ..., m, where M-j is an element of F-dxr, b(j) is an element of F-r, F is an element of {R, C} and we call it generalized affine phase retrieval. We first develop a framework for generalized affine phase retrieval with presenting several necessary and sufficient conditions for {(M-j, b(j))}(j=1)(m) having generalized affine phase retrieval property. Next, we focus on the minimal measurement number problem and establish some results for it. Particularly, we show if {(M-j, b(j))}(j=1)(m), subset of F-dxr x F-r has generalized affine phase retrieval property, then m >= d+ left perpendiculard/rright perpendicular for F = R (m >= 2d+ left perpendiculard/rright perpendicular for F = C). We also show that the lower bounds are tight provided r vertical bar d. These results imply that one can reduce the measurement number by raising r, i.e. the rank of M-j. This highlights a notable difference between generalized affine phase retrieval and generalized phase retrieval. Furthermore, using tools of algebraic geometry, we show that m >= 2d (resp. m >= 4d-1) generic measurements A = {(M-j, b(j))}(j=1)(m) have the generalized phase retrieval property for F = R (resp. F = C). (C) 2021 Elsevier Inc. All rights reserved.
关键词	Phase retrieval Determinant variety Matrix recovery
DOI	10.1016/j.aam.2021.102213
收录类别	SCI
语种	英语
资助项目	National Science Fund for Distinguished Young Scholars grant[12025108] ; Beijing Natural Science Foundation[Z180002] ; NSFC[12021001] ; NSFC[11688101]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000680040300009
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/58991
专题	中国科学院数学与系统科学研究院
通讯作者	Huang, Meng
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Comp Math, LSEC, Beijing 100091, Peoples R China 2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Huang, Meng,Xu, Zhiqiang. Phase retrieval from the norms of affine transformations[J]. ADVANCES IN APPLIED MATHEMATICS,2021,130:20.
APA	Huang, Meng,&Xu, Zhiqiang.(2021).Phase retrieval from the norms of affine transformations.ADVANCES IN APPLIED MATHEMATICS,130,20.
MLA	Huang, Meng,et al."Phase retrieval from the norms of affine transformations".ADVANCES IN APPLIED MATHEMATICS 130(2021):20.