Abstract

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

Citations