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Generalized phase retrieval: Measurement number, matrix recovery and beyond
Wang, Yang1; Xu, Zhiqiang2
AbstractIn 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 x*A(1)x, ..., x*A(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.
KeywordPhase retrieval Frames Fusion frames Fourier transform Measurement number Low rank matrix recovery Bilinear form Embedding
Funding ProjectHong 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 Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000477689000006
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Document Type期刊论文
Corresponding AuthorWang, Yang
Affiliation1.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
Recommended Citation
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.
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