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Unformatted text preview: that ∇f ∗ = (∇f )−1 and that f is closed.
72 7.21 Ellipsoidal peeling. In this problem, you will implement an outlier identiﬁcation technique using
L¨wnerJohn ellipsoids. Given a set of points D = {x1 , . . . , xN } in Rn , the goal is to identify a
o
set O ⊆ D that are anomalous in some sense. Roughly speaking, we think of an outlier as a point
that is far away from most of the points, so we would like the points in D \ O to be relatively close
together, and to be relatively far apart from the points in O.
We describe a heuristic technique for identifying O. We start with O = ∅ and ﬁnd the minimum
volume (L¨wnerJohn) ellipsoid E containing all xi ∈ O (which is all xi in the ﬁrst step). Each
o
/
iteration, we ﬂag (i.e., add to O) the point that corresponds to the largest dual variable for the
constraint xi ∈ E ; this point will be one of the points on the boundary of E , and intuitively, it will
be the one for whom the constraint is ‘most’ binding. We then plot vol E (on a log scale) versus
card O and hope that we see a sharp drop in the curv...
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 Fall '13
 F.Borrelli
 The Aeneid

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