Makalah : A multi two-spiral benchmark problem

Berikut sebagian isi  makalahnya :

Support Vector Machines (SVM) for solving pattern recognition and nonlinear function estimation problems have been introduced in [7]. The idea of SVM is mapping the training data nonlinearly into a higher-dimensional feature space, then construct a separating hyperplane with maximum margin there. This yields a nonlinear decision boundary in input space. By the use of a kernel function, either polynomial, splines, radial basis function (RBF) or multilayer perceptron, it is possible to compute the separating hyperplane without explicitly carrying out the map into the feature space. While classical Neural Networks techniques suffer from the existence of many local minima, SVM solutions are obtained from quadratic programming problems possessing a global solution.

Recently, least squares (LS) versions of SVM have been investigated for classification [5] and function estimation [6]. In these LS-SVM formulations one computes the solution by solving a linear system instead of quadratic programming. This is due to the use of equality instead of inequality constraints in the problem formulation. In [1, 4] such linear systems have been called Karush-Kuhn-Tucker (KKT) systems and their numerical stability has been investigated. This linear system can be efficiently solved by iterative methods such as conjugate gradient [2], and enables solving large scale classification problems. As an example we show the excellent performance on a multi two-spiral benchmark problem, which is known to be a difficult test case for neural network classifiers [3].

LEAST SQUARES SUPPORT VECTOR MACHINES

Given a training set of N data points , where is the k-th input pattern and is the k-th output pattern, the classifier can be constructed using the support vector method in the form

where  are called support values and b is a constant. The  is the kernel, which can be either  (linear SVM);  (polynomial SVM of degree d);  (multilayer perceptron SVM), or  (RBF SVM), where , and are constants.

For instance, the problem of classifying two classes is defined as

This can also be written as

where is a nonlinear function mapping of the input space to a higher dimensional space. LS-SVM classifiers\

subjects to the equality constraints

The Lagrangian is defined as

with Lagrange multipliers  (called support values).

The conditions for optimality are given by

for . After elimination of  and one obtains the solution

with and .  Mercer’s condition is applied to the matrix with

The kernel parameters, i.e. s for RBF kernel, can be optimally chosen by optimizing an upper bound on the VC dimension. The support values a_k are proportional to the errors at the data points in the LS-SVM case, while in the standard SVM case many support values are typically equal to zero. When solving large linear systems, it becomes needed to apply iterative methods [2].

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