Minimum-Volume Enclosing Ellipsoids and Core Sets

P. Kumar and E. A. Yildirim
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P. Kumar: Florida State University
E. A. Yildirim: Stony Brook University

Journal of Optimization Theory and Applications, 2005, vol. 126, issue 1, No 1, 21 pages

Abstract: Abstract We study the problem of computing a (1+ε)-approximation to the minimum-volume enclosing ellipsoid of a given point set $${\cal S} = \{p^{1}, p^{2}, \dots, p^{n}\} \subseteq {\mathbb R}^{d}$$ . Based on a simple, initial volume approximation method, we propose a modification of the Khachiyan first-order algorithm. Our analysis leads to a slightly improved complexity bound of $$O(nd^{3}/\epsilon)$$ operations for $$\epsilon \in(0, 1)$$ . As a byproduct, our algorithm returns a core set $${\cal X} \subseteq {\cal S}$$ with the property that the minimum-volume enclosing ellipsoid of $${\cal X}$$ provides a good approximation to that of $${\cal S}$$ . Furthermore, the size of $${\cal X}$$ depends on only the dimension d and ε, but not on the number of points n. In particular, our results imply that $$\vert {\cal X} \vert = O(d^{2}/\epsilon)$$ for $$\epsilon \in(0, 1)$$ .

Keywords: Löwner ellipsoids; core sets; approximation algorithms (search for similar items in EconPapers)
Date: 2005
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Citations: View citations in EconPapers (11)

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DOI: 10.1007/s10957-005-2653-6

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