The Aitchison and Aitken Kernel Function Revisited

Abstract

Over three decades ago Aitchison and Aitken proposed a novel kernel function for estimating the density functions of underlying distributions in discrete input spaces. To the best of our knowledge, it has not been shown whether this kernel function is positive definite ({\it i.e.}, a reproducing kernel function) on these spaces. Its positive definiteness would have enriched and enlarged its applicability domain: a positive definite kernel function has an associated Reproducing Kernel Hilbert Space, a framework on which a variety of powerful statistical and machine learning schemes can be developed. This paper aims to demonstrate that Aitchison and Aitken's kernel function is indeed positive definite on discrete metric spaces. We also touch on possible applications of the proposed theorem.