Uniform spaces and the Newtonian structure of (big)data affinity kernels

Abstract

Let $X$ be a (data) set. Let $K(x,y)>0$ be a measure of the affinity between the data points $x$ and $y$. We prove that $K$ has the structure of a Newtonian potential $K(x,y)=\varphi(d(x,y))$ with $\varphi$ decreasing and $d$ a quasi-metric on $X$ under two mild conditions on $K$. The first is that the affinity of each $x$ to itself is infinite and that for $x\neq y$ the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between $x$ and $y$ is larger than $\lambda>0$ and the affinity of $y$ and $z$ is also larger than $\lambda$, then the affinity between $x$ and $z$ is larger than $\nu(\lambda)$. The function $\nu$ is concave, increasing, continuous from $\mathbb{R}^+$ onto $\mathbb{R}^+$ with $\nu(\lambda)<\lambda$ for every $\lambda>0$.