Paper Info
Title
The max‐length‐vector line of best fit to a set of vector subspaces and an optimization problem over a set of hyperellipsoids
Abstract
Let C={V1,...,Vk} be a collection of subspaces of a finite-dimensional real vector space V. Let L denote a one-dimensional subspace of V, and let (L,V-i) denote the principal angle between L and V-i. Motivated by a problem in data analysis, we seek an L that maximizes the function F(L)=Sigma(i)cos(L,Vi). Conceptually, this is the line through the origin that best represents C with respect to the criterion F(L). A reformulation shows that L is spanned by a vector v=Sigma ivi, which maximizes the function G(v1,...,vk)=||Sigma ivi||2 subject to the constraints v(i)V(i) and ||v(i)||=1. In this setting, v is seen to be the longest vector that can be decomposed into unit vectors lying on prescribed hyperspheres. A closely related problem is to find the longest vector that can be decomposed into vectors lying on prescribed hyperellipsoids. Using Lagrange multipliers, the critical points of either problem can be cast as solutions of a multivariate eigenvalue problem. We employ homotopy continuation and numerical algebraic geometry to solve this problem and obtain the extremal decompositions. Copyright (c) 2015 John Wiley & Sons, Ltd.
Year
DOI
Venue
2015
10.1002/nla.1965
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Keywords
Field
DocType
numerical algebraic geometry,homotopy continuation,nonlinear optimization,principal angles,Grassmannian,multivariate eigenvalue problem
Mathematical optimization,Vector space,Combinatorics,Subspace topology,Lagrange multiplier,Linear subspace,Grassmannian,Optimization problem,Mathematics,Eigenvalues and eigenvectors,Unit vector
Journal
Volume
Issue
ISSN
22.0
3.0
1070-5325
Citations 
PageRank 
References 
0
0.34
19
Authors
5
Name
Order
Citations
PageRank
Daniel J. Bates110312.03
Brent Davis220.80
Michael Kirby313714.40
Justin Marks400.34
Chris Peterson5296.26