Learning the kernel based on error bounds

The problem of learning a kernel is considered based on minimizing the generalization bounds. According to the bounds, a bi-regularization criterion is developed for learning a kernel from the data. The relations between the criterion and some established criteria, such as kernel-target alignment and the regularization criterion, is discussed. Using the relations, we connect the kernel-target alignment and the generalization of kernel-based algorithms. Moreover, we consider the kernel-learning problem with the bi-regularization criterion when the kernel is in the convex hull of basic 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 n+1 basic kernels, where n is the sample size. And a saddle theorem is developed to characterize the optimal kernel.