Eigenvalues perturbation of integral operator for kernel selection

This paper introduces new kernel selection criteria based on the eigenvalues perturbation of the integral operator, and establishes the connection between eigen values perturbations and generalization error.

Abstract

Kernel selection is one of the key issues both in recent research and application of kernel methods. This is usually done by minimizing either an estimate of generalization error or some other related performance measure. It is well known that a kernel matrix can be interpreted as an empirical version of a continuous integral operator, and its eigenvalues converge to the eigenvalues of integral operator. In this paper, we introduce new kernel selection criteria based on the eigenvalues perturbation of the integral operator. This perturbation quantifies the difference between the eigenvalues of the kernel matrix and those of the integral operator. We establish the connection between eigenvalues perturbation and generalization error. By minimizing the derived generalization error bounds, we propose the kernel selection criteria. Therefore the kernel chosen by our proposed criteria can guarantee good generalization performance. To compute the values of our criteria, we present a method to obtain the eigenvalues of integral operator via the Fourier transform. Experiments on benchmark datasets demonstrate that our kernel selection criteria are sound and effective.