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)/ǫ) dimensions such that all pairwise distances are distorted by at most 1 + ǫ. We study the following basic question: Does there exist an analogue of the JohnsonLindenstrauss 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 l1 such that any linear embedding into l d 1 must incur a distortion of Ω( √ n/d). This bound is tight up to a logn 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 K2,3-free graphs, giving embeddings into l O(log n) 1 with constant distortion. Finally, we also present lower bounds on dimension reduction techniques for other lp 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 posi-