Dimension Reduction in the \ell _1 Norm

• Authors: Moses Charikar;Amit Sahai
• Venue: FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
• Year: 2002

The Johnson-Lindenstrauss Lemma shows that any set of n points in Euclidean space can be mapped linearly down to O0({{(\log n)} \mathord{\left/ {\vphantom {{(\log n)} {\varepsilon ^2 }}} \right. \kern-\nulldelimiterspace} {\varepsilon ^2 }}) dimensions such that all pairwise distances are distorted by at most 1 + \varepsilon. We study the following basic question: Does there exist an analogue of the Johnson-Lindenstrauss Lemma for the \ell _1 norm?Note that Johnson-Lindenstrauss Lemma gives a linear embedding which is independent of the point set. For the \ell _1 norm, we show that one cannot hope to use linear embeddingsas 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 n\ell _1^n such that any linear embedding into \ell _1^d must incur a distortion of \Omega (\sqrt {{n \mathord{\left/ {\vphantom {n d}} \right. \kern-\nulldelimiterspace} d}}). This bound is tight up to a log n factor. We then initiate a systematic study of general classes of \ell _1 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 \ell _1 withconstant distortion. Finally, we also present lower bounds on dimension reduction techniques for other \ell _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 \ell _1. 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.