Universal Kernels on Non-Standard Input Spaces

During the last years support vector machines (SVMs) have been successfully applied even in situations where the input space $X$ is not necessarily a subset of $R^d$. Examples include SVMs using probability measures to analyse e.g. histograms or coloured images, SVMs for text classification and web mining, and SVMs for applications from computational biology using, e.g., kernels for trees and graphs. Moreover, SVMs are known to be consistent to the Bayes risk, if either the input space is a complete separable metric space and the reproducing kernel Hilbert space (RKHS) $H\subset L_p(P_X)$ is dense, or if the SVM is based on a universal kernel $k$. So far, however, there are no RKHSs of practical interest known that satisfy these assumptions on $\cH$ or $k$ if $X \not\subset R^d$. We close this gap by providing a general technique based on Taylor-type kernels to explicitly construct universal kernels on compact metric spaces which are not subset of $R^d$. We apply this technique for the following special cases: universal kernels on the set of probability measures, universal kernels based on Fourier transforms, and universal kernels for signal processing.