An improved training algorithm for nonlinear kernel discriminants

A simple method to derive nonlinear discriminants is to map the samples into a high-dimensional feature space F using a nonlinear function and then to perform a linear discriminant analysis in F. Clearly, if F is a very high, or even infinitely, dimensional space, designing such a receiver may be a computationally intractable problem. However, using Mercer kernels, this problem can be solved without explicitly mapping the data to F. Recently, a powerful method of obtaining nonlinear kernel Fisher discriminants (KFDs) has been proposed, and very promising results were reported when compared with the other state-of-the-art classification techniques. In this paper, we present an extension of the KFD method that is also based on Mercer kernels. Our approach, which is called the nonlinear kernel second-order discriminant (KSOD), consists of determining a nonlinear receiver via optimization of a general form of second-order measures of performance. We also propose a complexity control procedure in order to improve the performance of these classifiers when few training data are available. Finally, simulations compare our approach with the KFD method.