An iterative algorithm for third-order tensor multi-rank minimization

doi:10.1007/s10589-015-9769-x

	An iterative algorithm for third-order tensor multi-rank minimization
	Yang, Lei 1; Huang, Zheng-Hai 1,2; Hu, Shenglong 1; Han, Jiye 3
刊名	COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
	2016
卷号	63 期号:1 页码:169-202
关键词	Third-order tensor multi-rank minimization Tensor completion Matrix rank minimization Operator splitting Convex relaxation
ISSN号	0926-6003
DOI	10.1007/s10589-015-9769-x
英文摘要	Recent work by Kilmer et al. (A third-order generalization of the matrix SVD as a product of third-order tensors, Department of Computer Science, Tufts University, Medford, MA, 2008; Linear Algebra Appl 435(3): 641-658, 2011; SIAM J Matrix Anal Appl 34(1): 148-172, 2013), and Braman (Linear Algebra Appl 433(7): 1241-1253, 2010) on tensor-tensor multiplication opens up a new avenue to study third-order tensors. Based on this new tensor-tensor multiplication and related concepts, some familiar tools of linear algebra can be extended to study third-order tensors. Motivated by this process, in this paper, we consider themulti-rank of a tensor as a sparsity measure and propose a new model, called third-order tensor multi-rank minimization, as an extension of matrix rank minimization. The operator splitting technique and the convex relaxation technique are used to tackle this problem. Based on these two powerful techniques, we propose a simple first-order and easy-to-implement algorithm to solve this problem. The proposed algorithm is shown to be globally convergent under some assumptions. The continuation technique is also applied to improve the numerical performance of the algorithm. Some preliminary numerical results demonstrate the efficiency of the proposed algorithm, and the potential value and applications of the multi-rank and the tensor multi-rank minimization model.
资助项目	National Nature Science Foundation of China[11171252] ; National Nature Science Foundation of China[11431002] ; National Nature Science Foundation of China[11201332]
WOS研究方向	Operations Research & Management Science ; Mathematics
语种	英语
出版者	SPRINGER
WOS记录号	WOS:000370560000008
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/22079]
专题	中国科学院数学与系统科学研究院
通讯作者	Huang, Zheng-Hai
作者单位	1.Tianjin Univ, Sch Sci, Dept Math, Tianjin 300072, Peoples R China 2.Tianjin Univ, Ctr Appl Math, Tianjin 300072, Peoples R China 3.Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Yang, Lei,Huang, Zheng-Hai,Hu, Shenglong,et al. An iterative algorithm for third-order tensor multi-rank minimization[J]. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS,2016,63(1):169-202.
APA	Yang, Lei,Huang, Zheng-Hai,Hu, Shenglong,&Han, Jiye.(2016).An iterative algorithm for third-order tensor multi-rank minimization.COMPUTATIONAL OPTIMIZATION AND APPLICATIONS,63(1),169-202.
MLA	Yang, Lei,et al."An iterative algorithm for third-order tensor multi-rank minimization".COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 63.1(2016):169-202.