Generalized Mercer theorem and its application to feature space related to indefinite kernels

The support vector machine (SVM) is well understood when kernel functions are positive definite. However, in practice, indefinite kernels arise and demand application in SVM. These indefinite kernels often yield good empirical classification results. However, they are hard to understand for missing geometrical and theoretical understanding. In this paper we focus our topic on the structure of feature space related to indefinite kernels. We develop a new method by improving Mercer theorem to construct the mapping that maps input data set into the high-dimensional feature space for indefinite kernels. Via this mapping, structure of the feature space is easily observed. By this, we obtain a sound framework and motivation for SVM with indefinite kernels.