Learning Convex Combinations of Continuously Parameterized Basic Kernels

We study the problem of learning a kernel which minimizes a regularization error functional such as that used in regularization networks or support vector machines. We consider this problem when the kernel is in the convex hull of basic kernels, for example, Gaussian kernels which are continuously parameterized by a compact set. We show that there always exists an optimal kernel which is the convex combination of at most m + 1 basic kernels, where m is the sample size, and provide a necessary and sufficient condition for a kernel to be optimal. The proof of our results is constructive and leads to a greedy algorithm for learning the kernel. We discuss the properties of this algorithm and present some preliminary numerical simulations.