Generalized phase retrieval: Measurement number, matrix recovery and beyond

doi:10.1016/j.acha.2017.09.003

	Generalized phase retrieval: Measurement number, matrix recovery and beyond
	Wang, Yang 1; Xu, Zhiqiang2
	2019-09-01
发表期刊	APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
ISSN	1063-5203
卷号	47 期号:2 页码:423-446
摘要	In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector x in R-d or C-d through quadratic samples xA(1)x, ..., xA(N)x. The generalized phase retrieval includes as special cases the standard phase retrieval as well as the phase retrieval by orthogonal projections. We first explore the connections among generalized phase retrieval, low-rank matrix recovery and nonsingular bilinear form. Motivated by the connections, we present results on the minimal measurement number needed for recovering a matrix that lies in a set W is an element of C-dxd. Applying the results to phase retrieval, we show that generic d x d matrices A(1), ..., A(N) have the phase retrieval property if N >= 2d - 1 in the real case and N >= 4d - 4 in the complex case for very general classes of A(1), ..., A(N), e.g. matrices with prescribed ranks or orthogonal projections. We also give lower bounds on the minimal measurement number required for generalized phase retrieval. For several classes of dimensions d we obtain the precise values of the minimal measurement number. Our work unifies and enhances results from the standard phase retrieval, phase retrieval by projections and low-rank matrix recovery. (C) 2017 Elsevier Inc. All rights reserved.
关键词	Phase retrieval Frames Fusion frames Fourier transform Measurement number Low rank matrix recovery Bilinear form Embedding
DOI	10.1016/j.acha.2017.09.003
语种	英语
资助项目	Hong Kong Research Grant Council[16306415] ; Hong Kong Research Grant Council[16317416] ; NSFC[11171336] ; NSFC[11422113] ; 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:000477689000006
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/35304
专题	计算数学与科学工程计算研究所
通讯作者	Wang, Yang
作者单位	1.Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Clear Water Bay, Hong Kong, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Comp Math, LSEC, Beijing 100091, Peoples R China
推荐引用方式 GB/T 7714	Wang, Yang,Xu, Zhiqiang. Generalized phase retrieval: Measurement number, matrix recovery and beyond[J]. APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS,2019,47(2):423-446.
APA	Wang, Yang,&Xu, Zhiqiang.(2019).Generalized phase retrieval: Measurement number, matrix recovery and beyond.APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS,47(2),423-446.
MLA	Wang, Yang,et al."Generalized phase retrieval: Measurement number, matrix recovery and beyond".APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 47.2(2019):423-446.