EXACT PENALTY FUNCTION FOR \ell2,1 NORM MINIMIZATION OVER THE STIEFEL MANIFOLD

doi:10.1137/20M1354313

CSpace

	EXACT PENALTY FUNCTION FOR \ell2,1 NORM MINIMIZATION OVER THE STIEFEL MANIFOLD
	Xiao, Nachuan 1,2; Liu, Xin 1,2; Ya-Xiang Yuan 1
	2021
发表期刊	SIAM JOURNAL ON OPTIMIZATION
ISSN	1052-6234
卷号	31 期号:4 页码:3097-3126
摘要	\ell 2,1 norm minimization with orthogonality constraints, which comprise a feasible region called the Stiefel manifold, has wide applications in statistics and data science. The state-ofthe-art approaches adopt a proximal gradient technique on either the Stiefel manifold or its tangent spaces. The consequent subproblem does not have a closed-form solution and hence requires an iterative procedure to solve, which is usually time-consuming. In this paper, we discover that the Lagrangian multipliers of the orthogonality constraints in this class of problems are of closed-form expressions. By using this closed-form expression, we introduce a penalty function for this type of problem. We theoretically demonstrate the equivalence between the penalty function and the original \ell 2,1 norm minimization under mild assumptions. Based on the exact penalty function, we propose an inexact proximal gradient method in which the subproblem is of closed-form solution. The global convergence and the worst case complexity are established. Numerical experiments illustrate the advantages of our method when compared with the existing proximal-based first-order methods.
关键词	orthogonality constraint Stiefel manifold nonsmooth optimization augmented Lagrangian method
DOI	10.1137/20M1354313
收录类别	SCI
语种	英语
资助项目	National Natural Science Foundation of China[11971466] ; National Natural Science Foundation of China[11991021] ; National Natural Science Foundation of China[11991020] ; Key Research Program of Frontier Sciences, Chinese Academy of Sciences[ZDBS-LY-7022] ; National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences ; Youth Innovation Promotion Association, Chinese Academy of Sciences ; NSFC[11688101]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000738355700024
出版者	SIAM PUBLICATIONS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/59836
专题	中国科学院数学与系统科学研究院
通讯作者	Xiao, Nachuan
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China 2.Univ Chinese Acad Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Xiao, Nachuan,Liu, Xin,Ya-Xiang Yuan. EXACT PENALTY FUNCTION FOR \ell2,1 NORM MINIMIZATION OVER THE STIEFEL MANIFOLD[J]. SIAM JOURNAL ON OPTIMIZATION,2021,31(4):3097-3126.
APA	Xiao, Nachuan,Liu, Xin,&Ya-Xiang Yuan.(2021).EXACT PENALTY FUNCTION FOR \ell2,1 NORM MINIMIZATION OVER THE STIEFEL MANIFOLD.SIAM JOURNAL ON OPTIMIZATION,31(4),3097-3126.
MLA	Xiao, Nachuan,et al."EXACT PENALTY FUNCTION FOR \ell2,1 NORM MINIMIZATION OVER THE STIEFEL MANIFOLD".SIAM JOURNAL ON OPTIMIZATION 31.4(2021):3097-3126.