KMS Of Academy of mathematics and systems sciences, CAS
SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM | |
Huang, Meng1; Xu, Zhiqiang | |
2020 | |
Source Publication | JOURNAL OF COMPUTATIONAL MATHEMATICS |
ISSN | 0254-9409 |
Volume | 38Issue:4Pages:638-660 |
Abstract | 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. |
Keyword | Low-rank matrix recovery Non-convex optimization Phase retrieval |
DOI | 10.4208/jcm.1902-m2018-0109 |
Indexed By | SCI |
Language | 英语 |
Funding Project | NSFC[91630203] ; NSFC[11688101] ; Youth Innovation Promotion Association CAS, Beijing Natural Science Foundation[Z180002] ; National Basic Research Programof China (973 Program)[2015CB856000] |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied ; Mathematics |
WOS ID | WOS:000526517900006 |
Publisher | GLOBAL SCIENCE PRESS |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/51152 |
Collection | 中国科学院数学与系统科学研究院 |
Corresponding Author | Huang, Meng |
Affiliation | 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 |
Recommended Citation 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. |
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