The max‐length‐vector line of best fit to a set of vector subspaces and an optimization problem over a set of hyperellipsoids. Issue 3 (12th January 2015) - Bates, Daniel J.; Davis, Brent R.; Kirby, Michael; Marks, Justin; Peterson, Chris
Summary Let <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3w65" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />C={V1, …, Vk}</alternatives> 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3w9q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />F(L)=∑icosθ(L, Vi)</alternatives>. Conceptually, this is the line through the origin that best represents <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3wd8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />C</alternatives> with respect to the criterion F(L). A reformulation shows that L is spanned by a vector <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3wht" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />v=∑ivi</alternatives>, which maximizes the function <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3vw1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />G(v1, …, vk)=||∑ivi||2</alternatives> 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.