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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 tradeoﬀ 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.
 Fall '13
 F.Borrelli
 The Aeneid

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