2 research outputs found
On -Support Vector Machines and Multidimensional Kernels
In this paper, we extend the methodology developed for Support Vector
Machines (SVM) using -norm (-SVM) to the more general case of
-norms with (-SVM). The resulting primal and dual
problems are formulated as mathematical programming problems; namely, in the
primal case, as a second order cone optimization problem and in the dual case,
as a polynomial optimization problem involving homogeneous polynomials.
Scalability of the primal problem is obtained via general transformations based
on the expansion of functionals in Schauder spaces. The concept of Kernel
function, widely applied in -SVM, is extended to the more general case
by defining a new operator called multidimensional Kernel. This object gives
rise to reformulations of dual problems, in a transformed space of the original
data, which are solved by a moment-sdp based approach. The results of some
computational experiments on real-world datasets are presented showing rather
good behavior in terms of standard indicators such a \textit{accuracy index}
and its ability to classify new data.Comment: 27 paes, 2 Figures, 2 table
Optimal arrangements of hyperplanes for multiclass classification
In this paper, we present a novel approach to construct multiclass
classifiers by means of arrangements of hyperplanes. We propose different mixed
integer (linear and non linear) programming formulations for the problem using
extensions of widely used measures for misclassifying observations where the
\textit{kernel trick} can be adapted to be applicable. Some dimensionality
reductions and variable fixing strategies are also developed for these models.
An extensive battery of experiments has been run which reveal the powerfulness
of our proposal as compared with other previously proposed methodologies.Comment: 8 Figures, 2 Table