Dimension reduction in the ℓ1 norm

The Johnson-Lindenstrauss lemma shows that any set of n points in Euclidean space can be mapped linearly down to O((log n)/ε²) dimensions such that all pairwise distances are distorted by at most 1+ε. We study the basic question of whether there exists an analogue of the Johnson-Lindenstrauss lemma for the ℓ₁ norm? Note that Johnson-Lindenstrauss lemma gives a linear embedding which is independent of the point set. For the ℓ₁ norm, we show that one cannot hope to use linear embeddings as a dimensionality reduction tool for general point sets, even if the linear embedding is chosen as a function of the given point set. In particular, we construct a set of O(n) points in ℓ₁ⁿ such that any linear embedding into ℓ₁^d must incur a distortion of Ω√(n/d). This bound is tight up to a log n factor. We then initiate a systematic study of general classes of ℓ₁ embeddable metrics that admit low dimensional, small distortion embeddings. In particular, we show dimensionality reduction theorems for tree metrics, circular-decomposable metrics, and metrics supported on K_2,3-free graphs, giving embeddings into ℓ₁^{O(log(2) n)} with constant distortion. Finally, we also present lower bounds on dimension reduction techniques for other ℓ_p norms. Our work suggests that the notion of a stretch-limited embedding, where no distance is stretched by more than a factor d in any dimension, is important to the study of dimension reduction for ℓ₁. We use such stretch limited embeddings as a tool for proving lower bounds for dimension reduction and also as an algorithmic tool for proving positive results.