Almost Optimal Explicit Johnson-Lindenstrauss Transformations

The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms in high dimensional geometry. Most known constructions of linear embeddings that satisfy the Johnson-Lindenstrauss property involve randomness. We address the question of explicitly constructing such embedding families and provide a construction with an almost optimal use of randomness: For 0 < δ, ε < 1, we give an explicit generator G : {0, 1} → Rs×n for s = O(log(1/δ)/ε) such that for all w ∈ R, ‖w‖ = 1, Pr y∈u{0,1} [ |‖G(y)w‖ − 1| > ε ] ≤ δ, and seed-length r = O ( log(n/δ) · log ( log(n/δ) ε )) . In particular, for δ = 1/poly(n) and fixed ε > 0 we get seed-length O(log n log log n). Previous constructions required at least O(log n) random bits to get polynomially small error. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 183 (2010)