The Grassmannian of affine subspaces

doi:10.1007/s10208-020-09459-8

CSpace

	The Grassmannian of affine subspaces
	Lim, Lek-Heng 1; Wong, Ken Sze-Wai 2; Ye, Ke 3
	2020-05-27
发表期刊	FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
ISSN	1615-3375
页码	38
摘要	The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being 0-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furthermore, it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be readily determined. On the other hand, like the Euclidean space, the affine Grassmannian serves as a concrete computational platform on which various distances, metrics, probability densities may be explicitly defined and computed via numerical linear algebra. Moreover, many standard problems in machine learning and statistics-linear regression, errors-in-variables regression, principal components analysis, support vector machines, or more generally any problem that seeks linear relations among variables that either best represent them or separate them into components-may be naturally formulated as problems on the affine Grassmannian.
关键词	Affine Grassmannian Affine subspaces Schubert calculus homotopy and (co)homology Probability densities Distances and metrics Multivariate data analysis
DOI	10.1007/s10208-020-09459-8
收录类别	SCI
语种	英语
资助项目	DARPA[D15AP00109] ; DARPA[NSFIIS 1546413] ; DARPA[DMS1209136] ; NSFC[11801548] ; NSFC[11688101] ; National Key R&D Program of China[2018YFA0306702] ; DARPA Director's Fellowship ; Eckhardt Faculty Fund ; Hundred Talents Program of the Chinese Academy of Sciences ; Recruitment Program of Global Experts of China
WOS研究方向	Computer Science ; Mathematics
WOS类目	Computer Science, Theory & Methods ; Mathematics, Applied ; Mathematics
WOS记录号	WOS:000535870100001
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/51524
专题	中国科学院数学与系统科学研究院
通讯作者	Lim, Lek-Heng
作者单位	1.Univ Chicago, Dept Stat, Computat & Appl Math Initiat, Chicago, IL 60637 USA 2.Univ Chicago, Dept Stat, Chicago, IL 60637 USA 3.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Lim, Lek-Heng,Wong, Ken Sze-Wai,Ye, Ke. The Grassmannian of affine subspaces[J]. FOUNDATIONS OF COMPUTATIONAL MATHEMATICS,2020:38.
APA	Lim, Lek-Heng,Wong, Ken Sze-Wai,&Ye, Ke.(2020).The Grassmannian of affine subspaces.FOUNDATIONS OF COMPUTATIONAL MATHEMATICS,38.
MLA	Lim, Lek-Heng,et al."The Grassmannian of affine subspaces".FOUNDATIONS OF COMPUTATIONAL MATHEMATICS (2020):38.