%A O. L. mangasarian
%T Generalized Support Vector Machines
%D October 1998
%R 98-14
%I COMPUTER SCIENCES DEPARTMENT, UNIVERSITY OF WISCONSIN
%C MADISON, WI
%X
By setting apart the two functions of a support vector machine:
separation of points
by a nonlinear surface in the original space of patterns, and maximizing
the distance between separating planes in a higher dimensional space,
we are able to define indefinite, possibly discontinuous, kernels,
not necessarily inner product ones,
that generate highly nonlinear separating surfaces. Maximizing
the distance between the separating planes in the higher
dimensional space is surrogated by support vector suppression,
which is achieved by minimizing
any desired norm of support vector multipliers.
The norm may be one induced by the separation kernel if it happens
to be positive definite, or a Euclidean or a polyhedral norm.
The latter norm leads to a linear program whereas the former
norms lead to convex quadratic programs, all with an
arbitrary separation kernel. A standard support
vector machine can be recovered by using the same kernel
for separation and support vector suppression.
On a simple test example, all models perform equally well
when a positive definite kernel is used. When a negative
definite kernel is used, we are unable to solve the nonconvex
quadratic program associated with a conventional support
vector machine, while all other proposed models remain
convex and easily generate a surface that separates
all given points.