A subset of points in a metric space is said to resolve it if each point in
the space is uniquely characterized by its distance to each point in the
subset. In particular, resolving sets can be used to represent points in
abstract metric spaces as Euclidean vectors. Importantly, due to the triangle
inequality, points close by in the space are represented as vectors with
similar coordinates, which may find applications in classification problems of
symbolic objects under suitably chosen metrics. In this manuscript, we address
the resolvability of Jaccard spaces, i.e., metric spaces of the form
$(2^X,\text{Jac})$, where $2^X$ is the power set of a finite set $X$, and
$\text{Jac}$ is the Jaccard distance between subsets of $X$. Specifically, for
different $a,b\in 2^X$, $\text{Jac}(a,b)=\frac{|a\Delta b|}{|a\cup b|}$, where
$|\cdot|$ denotes size (i.e., cardinality) and $\Delta$ denotes the symmetric
difference of sets. We combine probabilistic and linear algebra arguments to
construct highly likely but nearly optimal (i.e., of minimal size) resolving
sets of $(2^X,\text{Jac})$. In particular, we show that the metric dimension of
$(2^X,\text{Jac})$, i.e., the minimum size of a resolving set of this space, is
$\Theta(|X|/\ln|X|)$.

该研究论文主要探讨 Jaccard 空间的可分解性，使用概率和线性代数方法构建了在该空间中具有高概率但几乎优化（即最小大小）的可分解集合，且该空间的度量维度为 θ(|X|/ln|X|)。