Embedding Probability Distributions into Low Dimensional ℓ_1: Tree Ising Models via Truncated Metrics

Given an arbitrary set of high dimensional points in ℓ_1, there are known negative results that preclude the possibility of always mapping them to a low dimensional ℓ_1 space while preserving distances with small multiplicative distortion. This is in stark contrast with dimension reduction in Euclidean space (ℓ_2) where such mappings are always possible. While the first non-trivial lower bounds for ℓ_1 dimension reduction were established almost 20 years ago, there has been limited progress in understanding what sets of points in ℓ_1 are conducive to a low-dimensional mapping. In this work, we study a new characterization of ℓ_1 metrics that are conducive to dimension reduction in ℓ_1. Our characterization focuses on metrics that are defined by the disagreement of binary variables over a probability distribution – any ℓ_1 metric can be represented in this form. We show that, for configurations of n points in ℓ_1 obtained from tree Ising models, we can reduce dimension to 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 ℓ_1 metric can be truncated with O(1) distortion and O(log(n)) blowup in dimension.