Regularization with Dot-Product Kernels

Alexander J. Smola, Zoltán L. Óvári, Robert C. Williamson

Abstract: In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) = k(x . y) satisfy Mer(cid:173) cer's condition and thus may be used in Support Vector Ma(cid:173) chines (SVM), Regularization Networks (RN) or Gaussian Pro(cid:173) cesses (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.