论文标题
$ l_1 $ - 瓦斯恒星指标的distortion：二维的故事
$L_1$-distortion of Wasserstein metrics: a tale of two dimensions
论文作者
论文摘要
通过分散基斯利科夫的论点，Naor和Schechtman证明了平面网格上的1-Wasserstein度量$ \ \ {0,1，\ dots n \}^2 $具有$ l_1 $ -distortial，下面由$ \ sqrt {\ log n} $的常数多个bouther键。我们提供了基斯拉科夫论点的新“维度”解释，表明，如果$ \ {g_n \} _ {n = 1}^\ infty $是一系列图，其等级维度和lipschitz-eppectral dimension和lipschitz-pectral dimensive an $ l_1 $ -distortion以下面的限制为$（\ log | g_n |）^{\ frac {1}δ} $的常数倍数。我们继续为$ \ oslash $ - 某些图形的能力计算这些尺寸。特别是，我们得到了钻石图的顺序$ \ {\ MathSf {d} _n \} _ {n = 1}^\ infty $具有等级维度，而Lipschitz-Spectral尺寸等于2，等于2，以$ \ n $ \ n $ \ n $ \ n $ __t $ \ d} $ __t;以下由$ \ sqrt {\ log |的常数倍数\ Mathsf {D} _n |} $。这回答了Dilworth，Kutzarova和Ostrovskii的一个问题，仅展示了$ L_1 $ - Embeddable图的第三个序列，其1-Wasserstein指标的顺序不是$ l_1 $ - embeddable。
By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We provide a new "dimensionality" interpretation of Kislyakov's argument, showing that, if $\{G_n\}_{n=1}^\infty$ is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $δ\in [2,\infty)$, then the 1-Wasserstein metric over $G_n$ has $L_1$-distortion bounded below by a constant multiple of $(\log |G_n|)^{\frac{1}δ}$. We proceed to compute these dimensions for $\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{\mathsf{D}_n\}_{n=1}^\infty$ has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over $\mathsf{D}_n$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log| \mathsf{D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable.
