Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Randomness-optimal oblivious sampling
Proceedings of the workshop on Randomized algorithms and computation
The space complexity of approximating the frequency moments
Journal of Computer and System Sciences
Algorithmic derandomization via complexity theory
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Derandomized dimensionality reduction with applications
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
Journal of Computer and System Sciences - Special issu on PODS 2001
On variants of the Johnson–Lindenstrauss lemma
Random Structures & Algorithms
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
The Fast Johnson-Lindenstrauss Transform and Approximate Nearest Neighbors
SIAM Journal on Computing
Explicit Dimension Reduction and Its Applications
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with sub-constant error
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Sparser Johnson-Lindenstrauss transforms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Explicit Dimension Reduction and Its Applications
SIAM Journal on Computing
Sparser Johnson-Lindenstrauss Transforms
Journal of the ACM (JACM)
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The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms. Constructions of linear embeddings satisfying the Johnson-Lindenstrauss property necessarily involve randomness and much attention has been given to obtain explicit constructions minimizing the number of random bits used. In this work we give explicit constructions with an almost optimal use of randomness: For 0 G : {0, 1}r → Rs×d for s = O(log(1/δ)/ε2) such that for all d-dimensional vectors w of norm one, Pry∈u{0,1}r [|||G(y)w||2 - 1| ε] ≤ δ, with seed-length r = O(log d + log(1/δ) ċ log (log(1/δ/ε)). In particular, for δ = 1/ poly(d) and fixed ε 0, we obtain seed-length O((log d)(log log d)). Previous constructions required Ω(log2 d) random bits to obtain polynomially small error. We also give a new elementary proof of the optimality of the JL lemma showing a lower bound of Ω(log(1/δ)/ε2) on the embedding dimension. Previously, Jayram and Woodruff [10] used communication complexity techniques to show a similar bound.