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)

• Record Type: Journal Article
• Authors: Bates, Daniel J.; Davis, Brent R.; Kirby, Michael; Marks, Justin; Peterson, Chris
• Is Part Of: Numerical linear algebra with applications. Volume 22:Issue 3(2015:May)
• Page Start: 453
• Page End: 464
• Publication Date: 2015-01-12
• Subjects: Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.1965
• Languages: English
• ISSNs: 1070-5325

Abstract

<abstract abstract-type="main" id="nla1965-abs-0001"> <title>Summary</title> <p id="nla1965-para-0001">Let <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3w65" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:nla:media:nla1965:nla1965-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">C</mml:mi><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>, </mml:mo><mml:mo>…</mml:mo><mml:mo>, </mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:math></alternatives></inline-formula> be a collection of subspaces of a finite‐dimensional real vector space <italic>V</italic>. Let <italic>L</italic> denote a one‐dimensional subspace of <italic>V</italic>, and let <italic>θ</italic>(<italic>L</italic>, <italic>V</italic><sub><italic>i</italic></sub>) denote the principal angle between <italic>L</italic> and <italic>V</italic><sub><italic>i</italic></sub>. Motivated by a problem in data analysis, we seek an <italic>L</italic> that maximizes the function <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3w9q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:nla:media:nla1965:nla1965-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi><mml:mo>(</mml:mo><mml:mi>L</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:mi>cos</mml:mi><mml:mi>θ</mml:mi><mml:mo>(</mml:mo><mml:mi>L</mml:mi><mml:mo>, </mml:mo><mml:msub><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:math></alternatives></inline-formula>. Conceptually, this is the line through the origin that best represents <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3wd8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:nla:media:nla1965:nla1965-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">C</mml:mi></mml:math></alternatives></inline-formula> with respect to the criterion <italic>F</italic>(<italic>L</italic>). A reformulation shows that <italic>L</italic> is spanned by a vector <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3wht" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:nla:media:nla1965:nla1965-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, which maximizes the function <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjj6w3vw1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:nla:media:nla1965:nla1965-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>, </mml:mo><mml:mo>…</mml:mo><mml:mo>, </mml:mo><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> subject to the constraints <italic>v</italic><sub><italic>i</italic></sub>∈<italic>V</italic><sub><italic>i</italic></sub> and ||<italic>v</italic><sub><italic>i</italic></sub>||=1. In this setting, <italic>v</italic> 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.</p> </abstract>