In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) = k(x ċ y) satisfy Mercer's condition and thus may be used in Support Vector Machines (SVM), Regularization Networks (RN) or Gaussian Processes (GP). In particular, we show that if the kernel is analytic (i.e. can be expanded in a Taylor series), all expansion coefficients have to be nonnegative. We give an explicit functional form for the feature map by calculating its eigenfunctions and eigenvalues.
['Alexander J. Smola', 'Zoltán L. Óvári', 'Robert C. Williamson']
Regularization with Dot-Product Kernels