关于度量嵌入的图局部版本
On graphically local versions of metric embeddings
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中文总结 AI 辅助
该研究针对图局部度量嵌入问题，通过通用归约证明其难度不低于全距离嵌入，给出最大度3图的嵌入下界，且明确该度条件最优。
中文摘要 AI 辅助
我们研究图局部度量嵌入问题，即把任意有限度量空间中的点嵌入到目标度量空间中，仅保留由有界度图G指定的部分 pairwise 距离，且失真较小。我们给出一个通用归约，表明在许多情况下，该问题并不比近似保留所有 pairwise 距离的嵌入更容易。作为该通用归约的示例，我们证明存在一个含n个点的欧氏度量空间X，以及一个最大度为3的图G=(X,E)，使得任何仅保留E指定距离至相对误差(1+ε)的X到ℓ₂ᵐ的嵌入，必须满足m=Ω(ε⁻²log n)。我们的下界与Johnson-Lindenstrauss引理给出的近似保留所有 pairwise 距离的维度上界匹配；此前这类下界仅对非收缩嵌入（Schechtman-Shraibman，《离散与计算几何》，2009）已知。此外，图的最大度为3的条件是最优的：对于最大度为2的图G（或更一般地，树宽至多为2），任何度量空间都可G-等距嵌入到任意二维赋范空间中。
英文摘要
We consider the problem of graphically local metric embedding, i.e. embedding points from an arbitrary finite metric space into a target metric space while preserving, up to a small distortion, only a subset of the pairwise distances specified by a bounded degree graph $G$. We provide a general reduction showing that, in many cases, this is no easier than embedding the points while approximately preserving all pairwise distances. As an illustration of our general reduction, we show that there exists a Euclidean metric space $X$ on $n$ points along with a graph $G = (X,E)$ of maximum degree $3$ such that any embedding of $X$ into $\ell_2^m$ which only preserves distances specified by $E$ up to a relative error of $(1+\varepsilon)$ must satisfy $m = Ω(\varepsilon^{-2}\log n)$. Our lower bound matches the upper bound on the dimension coming from the Johnson-Lindenstrauss lemma for approximately preserving all pairwise distances; previously, such a lower bound was known only for the class of noncontracting embeddings [Schechtman-Shraibman, Discrete & Computational Geometry, 2009]. Moreover, the condition that the maximum degree of the graph is $3$ is best possible: for graphs $G$ of maximum degree $2$ (or more generally, treewidth at most $2$), any metric space embeds $G$-isometrically into any two-dimensional normed space.
