Spectral equivalence of unsymmetric kernel matrices and applications

Abstract

Symmetric kernel matrices are a well-researched topic in the literature of kernel based approximation. In particular stability properties in terms of lower bounds on the smallest eigenvalue of such symmetric kernel matrices are thoroughly investigated, as they play a fundamental role in theory and practice. In this work, we focus on unsymmetric kernel matrices and derive stability properties under small shifts by establishing a spectral equivalence to their unshifted, symmetric versions. This extends and generalizes results for translational invariant kernels upon Quak et al [SIAM Journal on Math. Analysis, 1993] and Sivakumar and Ward [Numerische Mathematik, 1993], however focussing instead on finitely smooth kernels. As applications, we consider convolutional kernels over domains, which are no longer translational invariant, but which are still an important class of kernels for applications. For these, we derive novel lower bounds for the smallest eigenvalue of the kernel matrices in terms of the separation distance of the data points, and thus derive stability bounds in terms of the condition number.