An elementary proof of the Johnson-Lindenstrauss Lemma (1999)
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@TECHREPORT{Dasgupta99anelementary,
author = {Sanjoy Dasgupta and Anupam Gupta},
title = {An elementary proof of the Johnson-Lindenstrauss Lemma},
institution = {},
year = {1999}
}
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Abstract
The Johnson-Lindenstrauss lemma shows that a set of n points in high dimensional Euclidean space can be mapped down into an O(log n=ffl 2 ) dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 \Sigma ffl). In this note, we prove this lemma using elementary probabilistic techniques. Computer Science Division, UC Berkeley. Email: dasgupta@cs.berkeley.edu. y Computer Science Division, UC Berkeley. Email: angup@cs.berkeley.edu. Supported by NSF grant CCR-9505448. 1 Introduction Johnson and Lindenstrauss [6] proved a fundamental result, which said that any n point set in Euclidean space could be embedded in O(log n=ffl 2 ) dimensions without distorting the distances between any pair of points by more than a factor of (1 \Sigma ffl), for any 0 ! ffl ! 1. Recently, this lemma has found several applications, including Lipschitz embeddings of graphs into normed spaces [7] and searching for approximate nearest neighbours [5]. The ori...
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