Given an arbitrary set of high dimensional points in $\ell_1$, there are known negative results that preclude the possibility of always 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 limited progress in understanding what sets of points in $\ell_1$ are conducive to a low-dimensional mapping. In this work, we study a new 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 truncated metrics which have been studied because of their applications in computer vision, and are objects of independent interest in metric geometry. Among other tools, we show how any $\ell_1$ metric can be truncated with $O(1)$ distortion and $O(\log(n))$ blowup in dimension.

翻译：给定ℓ₁空间中任意高维点集，已知存在负面结论：无法始终在保持较小乘法失真条件下将其映射到低维ℓ₁空间。这与欧氏空间（ℓ₂）中始终可行的降维形成鲜明对比。尽管ℓ₁降维的首批非平凡下界早在20年前就已确立，但关于哪些ℓ₁点集适合低维映射的理解仍进展有限。本文研究了有利于ℓ₁降维的ℓ₁度量的新特征。我们的特征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ℓ₁度量均可表示为该形式。研究表明，对于从树状伊辛模型获得的n个ℓ₁空间点的配置，我们可以在恒定失真条件下将维度降低至$\mathrm{polylog}(n)$。在此过程中，我们开发了截断度量嵌入的技术工具——这类度量因计算机视觉中的应用而受到研究，且本身是度量几何中具有独立价值的研究对象。除其他工具外，我们还证明了任何ℓ₁度量均可通过O(1)失真和O(log(n))维度膨胀实现截断。