This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 1 and C2 correctly, and we can maximize the size of the neighborhood. This problem can
be expressed as
maximize t
subject to aT x + b > 0 if dist(x, C1 ) ≤ t,
aT x + b < 0 if dist(x, C2 ) ≤ t, where dist(x, C ) = miny∈C x − y 2 .
This is illustrated in the ﬁgure. The centers of the shaded disks form the set C1 . The centers
of the other disks form the set C2 . The set of points at a distance less than t from Ci is the
union of disks with radius t and center in Ci . The hyperplane in the ﬁgure separates the two
expanded sets. We are interested in expanding the circles as much as possible, until the two
expanded sets are no longer separable by a hyperplane.
aT x + b < 0 aT x + b > 0 Since the constraints are homogeneous in a, b, we can again replace them with nonstrict
inequalities
maximize t
subject to aT x + b ≥ 1 if dist(x, C1 ) ≤ t,
(26)
aT x + b ≤ −1 if dist(x, C2 ) ≤ t.
61 The variables are a, b, and t.
(b) Next we consider an extension to more than two classes. If m > 2 we can use a decision
function
f (x) = argmax (aT x + bi ),
i
i=1,...,m n parameterized by m vectors ai ∈ R and m scalars bi . To ﬁnd f , we...
View
Full
Document
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

Click to edit the document details