Given an arbitrary set of high dimensional points in $\ell_1$, there are known negative results that preclude the possibility of mapping them to a low dimensional $\ell_1$ space while preserving distances with small multiplicative distortion. This is in stark contrast with dimension reduction in Euclidean space ($\ell_2$) where such mappings are always possible. While the first non-trivial lower bounds for $\ell_1$ dimension reduction were established almost 20 years ago, there has been minimal progress in understanding what sets of points in $\ell_1$ are conducive to a low-dimensional mapping. In this work, we shift the focus from the worst-case setting and initiate the study of a characterization of $\ell_1$ metrics that are conducive to dimension reduction in $\ell_1$. Our characterization focuses on metrics that are defined by the disagreement of binary variables over a probability distribution -- any $\ell_1$ metric can be represented in this form. We show that, for configurations of $n$ points in $\ell_1$ obtained from tree Ising models, we can reduce dimension to $\mathrm{polylog}(n)$ with constant distortion. In doing so, we develop technical tools for embedding capped metrics (also known as truncated metrics) which have been studied because of their applications in computer vision, and are objects of independent interest in metric geometry.

翻译：给定 $\ell_1$ 空间中任意一组高维点集，已知的否定结果排除了将其映射到低维 $\ell_1$ 空间并保持距离具有小乘法失真的可能性。这与欧几里得空间 ($\ell_2$) 中的维度约简形成鲜明对比，后者中此类映射总是可能的。尽管 $\ell_1$ 维度约简的首个非平凡下界在近20年前就已建立，但关于理解 $\ell_1$ 空间中哪些点集适合低维映射的研究进展甚微。本研究将关注点从最坏情况设定转移，首次系统刻画了 $\ell_1$ 度量中适合 $\ell_1$ 维度约简的特征。我们的刻画聚焦于由二元变量在概率分布上的不一致性所定义的度量——任何 $\ell_1$ 度量均可表示为该形式。我们证明，对于从树形伊辛模型获得的 $\ell_1$ 空间中 $n$ 个点的配置，可以在恒定失真条件下将维度约简至 $\mathrm{polylog}(n)$。在此过程中，我们开发了用于嵌入有界度量（也称为截断度量）的技术工具，这类度量因在计算机视觉中的应用而备受关注，且其本身在度量几何中具有独立研究价值。