  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.
