Classes of kernels for machine learning: A statistics perspective

In this paper, we present classes of kernels for machine learning from a statistics perspective. Indeed, kernels are positive definite functions and thus also covariances. After discussing key properties of kernels, as well as a new formula to construct kernels, we present several important classes of kernels: anisotropic stationary kernels, isotropic stationary kernels, compactly supported kernels, locally stationary kernels, nonstationary kernels, and separable nonstationary kernels. Compactly supported kernels and separable nonstationary kernels are of prime interest because they provide a computational reduction for kernel-based methods. We describe the spectral representation of the various classes of kernels and conclude with a discussion on the characterization of nonlinear maps that reduce nonstationary kernels to either stationarity or local stationarity.