The max‐length‐vector line of best fit to a set of vector subspaces and an optimization problem over a set of hyperellipsoids |
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| Authors: | Daniel J. Bates Brent R. Davis Michael Kirby Justin Marks Chris Peterson |
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| Affiliation: | 1. Department of Mathematics, Colorado State University, Fort Collins, CO, USA;2. Department of Mathematics, Bowdoin College, Brunswick, ME, USA |
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| Abstract: | Let  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  . Conceptually, this is the line through the origin that best represents  with respect to the criterion F(L). A reformulation shows that L is spanned by a vector  , which maximizes the function  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 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 © 2015 John Wiley & Sons, Ltd. |
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| Keywords: | numerical algebraic geometry homotopy continuation nonlinear optimization principal angles Grassmannian multivariate eigenvalue problem |
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