标题： 均匀空间与（大数据）相似性核的牛顿结构 显示英文标题

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

摘要： 设$X$为一个（数据）集。 设$K(x,y)>0$为数据点$x$和$y$之间的亲和力的度量。 我们证明了在对$K$的两个温和条件下，$K$具有牛顿势$K(x,y)=\varphi(d(x,y))$的结构，其中$\varphi$是递减的，$d$是$X$上的拟度量。 第一个是每个$x$与自身的亲和力是无限的，而$x\neq y$的亲和力是正的且有限的。 第二个是定量传递性；如果$x$和$y$之间的亲和力大于$\lambda>0$，且$y$和$z$的亲和力也大于$\lambda$，则$x$和$z$之间的亲和力大于$\nu(\lambda)$。 函数$\nu$是凹函数，从$\mathbb{R}^+$到$\mathbb{R}^+$单调递增且连续，对于每个$\lambda>0$都有$\nu(\lambda)<\lambda$。 显示英文摘要

摘要： 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 $\lambda>0$.