SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM

doi:10.4208/jcm.1902-m2018-0109

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	SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM
	Huang, Meng 1; Xu, Zhiqiang
	2020
发表期刊	JOURNAL OF COMPUTATIONAL MATHEMATICS
ISSN	0254-9409
卷号	38 期号:4 页码:638-660
摘要	We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X is an element of R-nxr from m scalar measurements y(i) = a(i)(inverted perpendicular) XX inverted perpendicular a(i), a(i) is an element of R-n, i = 1, ... , m. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential-type gradient descent algorithm, to minimize a non-convex objective function f(U) = 1/4m Sigma(m)(i=1)(y(i) - a(i)(inverted perpendicular) UU inverted perpendicular a(i))(2). This algorithm starts with a careful initialization, and then refines this initial guess by iteratively applying exponential-type gradient descent. Particularly, we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr), and our iteration algorithm can converge linearly to the true X (up to an orthogonal matrix) with m = O (nr log(cr)) Gaussian random measurements.
关键词	Low-rank matrix recovery Non-convex optimization Phase retrieval
DOI	10.4208/jcm.1902-m2018-0109
收录类别	SCI
语种	英语
资助项目	NSFC[91630203] ; NSFC[11688101] ; Youth Innovation Promotion Association CAS, Beijing Natural Science Foundation[Z180002] ; National Basic Research Programof China (973 Program)[2015CB856000]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000526517900006
出版者	GLOBAL SCIENCE PRESS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/51152
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 100190, 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. SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM[J]. JOURNAL OF COMPUTATIONAL MATHEMATICS,2020,38(4):638-660.
APA	Huang, Meng,&Xu, Zhiqiang.(2020).SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM.JOURNAL OF COMPUTATIONAL MATHEMATICS,38(4),638-660.
MLA	Huang, Meng,et al."SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM".JOURNAL OF COMPUTATIONAL MATHEMATICS 38.4(2020):638-660.