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

Content Provider	IEEE Xplore Digital Library
Author	Charikar, M. Sahai, A.
Copyright Year	2002
Description	Author affiliation: Princeton Univ., NJ, USA (Charikar, M.; Sahai, A.)
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.
Sponsorship	IEEE Comput. Soc. Tech. Committee on Maht. Found. Comput. (TCMF) ACM/SIGACT
Starting Page	551
Ending Page	560
File Size	806397
Page Count	10
File Format	PDF
ISBN	0769518222
ISSN	02725428
DOI	10.1109/SFCS.2002.1181979
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2002-11-19
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Extraterrestrial measurements Upper bound Books Embedded computing
Content Type	Text
Resource Type	Article

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