Extending Mercer's expansion to indefinite and asymmetric kernels

TL;DR

This paper extends Mercer's expansion to continuous, indefinite, and asymmetric kernels, providing theoretical foundations, convergence conditions, and an algorithm for practical computation.

Contribution

It offers a rigorous extension of Mercer's theorem to a broader class of kernels, including indefinite and asymmetric ones, with convergence analysis and computational methods.

Findings

Mercer's expansion may not be pointwise convergent for indefinite kernels

Under bounded variation, the expansion converges almost everywhere and unconditionally

An algorithm for computing Mercer's expansion for general kernels is proposed

Abstract

Mercer's expansion and Mercer's theorem are cornerstone results in kernel theory. While the classical Mercer's theorem only considers continuous symmetric positive definite kernels, analogous expansions are effective in practice for indefinite and asymmetric kernels. In this paper we extend Mercer's expansion to continuous kernels, providing a rigorous theoretical underpinning for indefinite and asymmetric kernels. We begin by demonstrating that Mercer's expansion may not be pointwise convergent for continuous indefinite kernels, before proving that the expansion of continuous kernels with bounded variation uniformly in each variable separably converges pointwise almost everywhere, almost uniformly, and unconditionally almost everywhere. We also describe an algorithm for computing Mercer's expansion for general kernels and give new decay bounds on its terms.