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Abstract:
We show that the recently proposed variant of the Support
Vector machine (SVM) algorithm, known as
-SVM, can be interpreted as a maximal separation between subsets of
the convex hulls of the data, which we call soft convex hulls. The
soft convex hulls are controlled by choice of the parameter
. If the intersection of the convex hulls is empty, the hyperplane
is positioned halfway between them such that the distance between
convex hulls, measured along the normal, is maximized; and if it is
not, the hyperplane's normal is similarly determined by the soft
convex hulls, but its position (perpendicular distance from the
origin) is adjusted to minimize the error sum. The proposed
geometric interpretation of
-SVM also leads to necessary and sufficient conditions for the
existence of a choice of
for which the
-SVM solution is nontrivial.
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