The Estimation Performance of Nonlinear Least Squares for Phase Retrieval

doi:10.1109/TIT.2020.2983562

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	The Estimation Performance of Nonlinear Least Squares for Phase Retrieval
	Huang, Meng 1,2; Xu, Zhiqiang 1,2
	2020-12-01
发表期刊	IEEE TRANSACTIONS ON INFORMATION THEORY
ISSN	0018-9448
卷号	66 期号:12 页码:7967-7977
摘要	Suppose that y = vertical bar Ax(0)vertical bar + eta where x(0) is an element of R-d is the target signal and eta is an element of R-m is a noise vector. The aim of phase retrieval is to estimate x(0) from y. A popular model for estimating x(0) is the nonlinear least squares (x) over cap := argmin(x) parallel to vertical bar Ax vertical bar - y parallel to 2. One has already developed many efficient algorithms for solving the model, such as the seminal error reduction algorithm. In this paper, we present the estimation performance of the model with proving that parallel to(x)over cap> - x(0)parallel to less than or similar to parallel to eta parallel to 2/root m under the assumption of A being a Gaussian random matrix. We also prove the reconstruction error parallel to eta parallel to(2)/root m is sharp. For the case where x(0) is sparse, we study the estimation performance of both the nonlinear Lasso of phase retrieval and its unconstrained version. Our results are non-asymptotic, and we do not assume any distribution on the noise eta. To the best of our knowledge, our results represent the first theoretical guarantee for the nonlinear least squares and for the nonlinear Lasso of phase retrieval.
关键词	Phase retrieval estimation performance nonlinear least squares nonlinear Lasso
DOI	10.1109/TIT.2020.2983562
收录类别	SCI
语种	英语
资助项目	NSFC[91630203] ; NSFC[11688101] ; Beijing Natural Science Foundation[Z180002]
WOS研究方向	Computer Science ; Engineering
WOS类目	Computer Science, Information Systems ; Engineering, Electrical & Electronic
WOS记录号	WOS:000594905600043
出版者	IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/57844
专题	中国科学院数学与系统科学研究院
通讯作者	Xu, Zhiqiang
作者单位	1.Univ Chinese Acad Sci, Sch Math Sci, Inst Computat Math, Beijing 100049, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Huang, Meng,Xu, Zhiqiang. The Estimation Performance of Nonlinear Least Squares for Phase Retrieval[J]. IEEE TRANSACTIONS ON INFORMATION THEORY,2020,66(12):7967-7977.
APA	Huang, Meng,&Xu, Zhiqiang.(2020).The Estimation Performance of Nonlinear Least Squares for Phase Retrieval.IEEE TRANSACTIONS ON INFORMATION THEORY,66(12),7967-7977.
MLA	Huang, Meng,et al."The Estimation Performance of Nonlinear Least Squares for Phase Retrieval".IEEE TRANSACTIONS ON INFORMATION THEORY 66.12(2020):7967-7977.