Dimension reduction in the /spl lscr//sub 1/ norm

Abstract

The Johnson-Lindenstrauss lemma shows that any set of n points in Euclidean space can be mapped linearly down to O((log n)//spl epsi//sup 2/) dimensions such that all pairwise distances are distorted by at most 1+/spl epsi/. We study the basic question of whether there exists an analogue of the Johnson-Lindenstrauss lemma for the /spl lscr//sub 1/ norm? Note that Johnson-Lindenstrauss lemma gives a linear embedding which is independent of the point set. For the /spl lscr//sub 1/ 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 /spl lscr//sub 1//sup n/ such that any linear embedding into /spl lscr//sub 1//sup d/ must incur a distortion of /spl Omega//spl radic/(n/d). This bound is tight up to a log n factor. We then initiate a systematic study of general classes of /spl lscr//sub 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 K/sub 2,3/-free graphs, giving embeddings into /spl lscr//sub 1//sup O(log(2) n)/ with constant distortion. Finally, we also present lower bounds on dimension reduction techniques for other /spl lscr//sub 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 /spl lscr//sub 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.