Topology of tensor ranks

doi:10.1016/j.aim.2020.107128

CSpace

	Topology of tensor ranks
	Comon, Pierre 1; Lim, Lek-Heng 2; Qi, Yang 3,4; Ye, Ke 5
	2020-06-24
发表期刊	ADVANCES IN MATHEMATICS
ISSN	0001-8708
卷号	367 页码:46
摘要	We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over C, the set of rank-rtensors and the set of symmetric rank-rsymmetric tensors are both path-connected if r is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over C. Over R, the set of rank-rtensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-rsymmetric d-tensors depends on the order d: connected when d is odd but not when d is even. Border rank and symmetric border rank over R have essentially the same path-connectedness properties as rank and symmetric rank over R. When ris greater than the complex generic rank, we are unable to discern any general pattern: For example, we show that border-rank-three tensors in R-2 circle times R-2 circle times R-2 fall into four connected components. For multilinear rank, the manifold of d-tensors of multilinear rank (r(1),..., r(d)) in C-n1 circle times center dot center dot center dot circle times C-nd is always path-connected, and the same is true in R-n1 circle times center dot center dot center dot circle times R-nd unless n(i) = r(i) = Pi(j not equal i) r(j) for some i is an element of{1,..., d}. Beyond path-connectedness, we determine, over both Rand C, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank. (C) 2020 Elsevier Inc. All rights reserved.
关键词	Border rank Symmetric rank Multilinear rank Path-connectedness Homotopy groups Bott periodicity
DOI	10.1016/j.aim.2020.107128
收录类别	SCI
语种	英语
资助项目	ERC[320594] ; DARPA[D15AP00109] ; NSF[IIS 1546413] ; NSF[DMS 1209136] ; NSF[DMS 1057064] ; DARPA Director's Fellowship ; NSFC[11801548] ; NSFC[11688101] ; National Key R&D Program of China[2018YFA0306702] ; Hundred Talents Program of the Chinese Academy of Sciences ; recruitment program for young professionals of China
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000526414600013
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/51137
专题	中国科学院数学与系统科学研究院
通讯作者	Qi, Yang
作者单位	1.CNRS, GIPSA Lab, F-38402 St Martin Dheres, France 2.Univ Chicago, Dept Stat, Computat & Appl Math Initiat, Chicago, IL 60637 USA 3.Ecole Polytech, IP Paris, CNRS, INRIA Saclay Ile de France, F-91128 Palaiseau, France 4.Ecole Polytech, IP Paris, CNRS, CMAP, F-91128 Palaiseau, France 5.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Comon, Pierre,Lim, Lek-Heng,Qi, Yang,et al. Topology of tensor ranks[J]. ADVANCES IN MATHEMATICS,2020,367:46.
APA	Comon, Pierre,Lim, Lek-Heng,Qi, Yang,&Ye, Ke.(2020).Topology of tensor ranks.ADVANCES IN MATHEMATICS,367,46.
MLA	Comon, Pierre,et al."Topology of tensor ranks".ADVANCES IN MATHEMATICS 367(2020):46.