Kernel Functions and Reproducing Kernel Hilbert Spaces

This chapter discusses the fundamental and advanced relevant concepts on Mercer's kernels and reproducing kernel Hilbert spaces (RKHSs). The fundamental building block of the kernel learning theory is the kernel function, which provides an elegant framework to compare complex and nontrivial objects. After its introduction, the chapter reviews the concept of an RKHS and also the representer theorem. It also discusses the main properties on kernel functions and their construction, as well as the basic ideas to work with complex objects and reproducing spaces. The chapter then introduces in detail support vector regression (SVR) algorithm, as it will be widely used and modified for building many of the digital signal processing (DSP) algorithms with kernel methods. Kernel methods rely on the properties of kernel functions. The chapter concludes with some synthetic examples illustrating the concepts and tools presented.