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A sparse Johnson: Lindenstrauss transform
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The JohnsonLindenstrauss lemma asserts that an n-pointset in any Euclidean space can be mapped to a Euclidean space ofdimension k = O(ε-2 log n)so that all distances are preserved up to a multiplicative factorbetween 1 - ε and 1 + ε. Known proofs obtain such amapping as a linear map Rn ’Rk with a suitable random matrix. We givea simple and self-contained proof of a version of theJohnsonLindenstrauss lemma that subsumes a basic versions by Indykand Motwani and a version more suitable for efficient computationsdue to Achlioptas. (Another proof of this result, slightlydifferent but in a similar spirit, was given independently by Indykand Naor.) An even more general result was established by Klartagand Mendelson using considerably heavier machinery.Recently, Ailon and Chazelle showed, roughly speaking, that agood mapping can also be obtained by composing a suitable Fouriertransform with a linear mapping that has a sparse random matrixM; a mapping of this form can be evaluated very fast. Intheir result, the nonzero entries of M are normallydistributed. We show that the nonzero entries can be chosen asrandom ± 1, which further speeds up the computation. We alsodiscuss the case of embeddings into Rkwith the ℓ1 norm. © 2008 Wiley Periodicals,Inc. Random Struct. Alg., 2008