Find the thickness of the maximum thickness

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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...
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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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