A Convex Parametrization of a New Class of Universal Kernel Functions for use in Kernel Learning

We propose a new class of universal kernel functions which admit a linear parametrization using positive semidefinite matrices. These kernels are generalizations of the Sobolev kernel and are defined by piecewise-polynomial functions. The class of kernels is termed "tessellated" as the resulting discriminant is defined piecewise with hyper-rectangular domains whose corners are determined by the training data. The kernels have scalable complexity, but each instance is universal in the sense that its hypothesis space is dense in $L_2$. Using numerical testing, we show that for the soft margin SVM, this class can eliminate the need for Gaussian kernels. Furthermore, we demonstrate that when the ratio of the number of training data to features is high, this method will significantly outperform other kernel learning algorithms. Finally, to reduce the complexity associated with SDP-based kernel learning methods, we use a randomized basis for the positive matrices to integrate with existing multiple kernel learning algorithms such as SimpleMKL.