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 (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.

The Parzen Window method: In terms of two vectors and one matrix

“…As a concrete example, we use the widely utilized kernel function (albeit in cheminformatics [15,16] , and references therein) introduced by Aitchison and Aitken (AA–kernel) [17,18] , where 0.5 < λ < 1 and d ( x, x i ) denotes the number of components in which x and x i disagree. This dissimilarity measure d ( x, x i ) can be conveniently expressed as [4] In passing, the AA–kernel is basically a discrete analogue of an isotropic Gaussian kernel [17,18] .…”

Combinatorial Bayesian Optimization with Random Mapping Functions to Convex Polytopes

“…Furthermore, in this work, we will employ a baseline with Aitchison and Aitken kernel (AA-kernel) (Aitchison and Aitken, 1976), which is defined on a discrete space. This AA-kernel is symmetric and positive definite (Mussa, 2013), which satisfies the requirements of positivedefinite kernels (Schölkopf and Smola, 2002).…”