Upper and Lower Bounds for Matrix Discrepancy

doi:10.1007/s00041-022-09976-w

	Upper and Lower Bounds for Matrix Discrepancy
	Xie, Jiaxin 2; Xu, Zhiqiang 1,3; Zhu, Ziheng 1,3
刊名	JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
	2022-12-01
卷号	28 期号:6 页码:23
关键词	Matrix discrepancy Tight frame Interlacing polynomials Kadison-Singer problem
ISSN号	1069-5869
DOI	10.1007/s00041-022-09976-w
英文摘要	The aim of this paper is to study the matrix discrepancy problem. Assume that xi(1), ..., xi(n) are independent scalar random variables with finite support and u(1), ..., u(n) is an element of C-d. Let C-0 be the minimal constant for which the following holds: Disc(u(1)u(1),..., u(n)u(n) ;xi(1), ..., xi(n)) := min(epsilon 1 subset of S1, ..., epsilon n subset of Sn) parallel to Sigma(n)(i=1) E[xi(i)]u(i) u(i)* - Sigma(n)(i=1) epsilon(i)u(i)u(i)parallel to <= C-0.sigma, where sigma(2) = parallel to Sigma(n)(i=1) Var [xi(i)] (u(i)u(i))(2)parallel to and S-j denotes the support of xi(j), j = 1, ..., n. Motivated by the technology developed by Bownik, Casazza, Marcus, and Speegle [7], we prove C-0 <= 3. This improves Kyng, Luh and Song's method with which C-0 <= 4 [21]. For the case where {u(i)}(i=1)(n) subset of C-d is a unit-norm tight frame with n <= 2d - 1 and xi(1), ..., xi(n) are independent Rademacher random variables, we present the exact value of Disc (u(1)u(1), ..., u(n)u(n) ;xi(1), ..., xi(n)) = root n/d.sigma, which implies C-0 >= root 2.
资助项目	National Science Fund for Distinguished Young Scholars[12025108] ; NSFC[12001026] ; NSFC[12071019] ; NSFC[12021001]
WOS研究方向	Mathematics
语种	英语
出版者	SPRINGER BIRKHAUSER
WOS记录号	WOS:000870331200002
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/60737]
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, Beijing 100049, Peoples R China 2.Beihang Univ, Sch Math Sci, LMIB Minist Educ, Beijing 100191, Peoples R China 3.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	Xie, Jiaxin,Xu, Zhiqiang,Zhu, Ziheng. Upper and Lower Bounds for Matrix Discrepancy[J]. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS,2022,28(6):23.
APA	Xie, Jiaxin,Xu, Zhiqiang,&Zhu, Ziheng.(2022).Upper and Lower Bounds for Matrix Discrepancy.JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS,28(6),23.
MLA	Xie, Jiaxin,et al."Upper and Lower Bounds for Matrix Discrepancy".JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 28.6(2022):23.