A explain how to solve this problem using convex or

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Unformatted text preview: ot, with your fitted vector field. 7.13 Robust minimum volume covering ellipsoid. Suppose z is a point in Rn and E is an ellipsoid in Rn with center c. The Mahalanobis distance of the point to the ellipsoid center is defined as M (z, E ) = inf {t ≥ 0 | z ∈ c + t(E − c)}, which is the factor by which we need to scale the ellipsoid about its center so that z is on its boundary. We have z ∈ E if and only if M (z, E ) ≤ 1. We can use (M (z, E ) − 1)+ as a measure of the Mahalanobis distance of the point z to the ellipsoid E . Now we can describe the problem. We are given m points x1 , . . . , xm ∈ Rn . The goal is to find the optimal trade-off between the volume of the ellipsoid E and the total Mahalanobis distance of the points to the ellipsoid, i.e., m i=1 (M (z, E ) − 1)+ . Note that this can be considered a robust version of finding the smallest volume ellipsoid that covers a set of points, since here we allow one or more points to be outside the ellipsoid. (a) Explain how...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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