The max-length-vector line of best fit to a set of vector subspaces and an optimization problem over a set of hyperellipsoids
چکیده
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 θ(L, Vi) denote the principal angle between L and Vi. Motivated by a problem in data analysis, we seek an L that maximizes the function F (L) = ∑ i cos θ(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 = ∑ i vi which maximizes the function G(v1, . . . , vk) = || ∑ i vi|| 2 subject to the constraints vi ∈ Vi and ||vi|| = 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 corresponds to finding 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 the problem and obtain the extremal decompositions. Copyright c © 2013 John Wiley & Sons, Ltd.