Communications of the ACM
A sparse Johnson: Lindenstrauss transform
Proceedings of the forty-second ACM symposium on Theory of computing
Fast locality-sensitive hashing
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
Sparser Johnson-Lindenstrauss transforms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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
An almost optimal unrestricted fast Johnson-Lindenstrauss transform
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Explicit Dimension Reduction and Its Applications
SIAM Journal on Computing
An Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Optimal Bounds for Johnson-Lindenstrauss Transforms and Streaming Problems with Subconstant Error
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Sparsity lower bounds for dimensionality reducing maps
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Sparser Johnson-Lindenstrauss Transforms
Journal of the ACM (JACM)
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The Fast Johnson–Lindenstrauss Transform (FJLT) was recently discovered by Ailon and Chazelle as a novel technique for performing fast dimension reduction with small distortion from ℓ 2 d to ℓ 2 k in time O(max {dlog d,k 3}). For k in [Ω(log d),O(d 1/2)], this beats time O(dk) achieved by naive multiplication by random dense matrices, an approach followed by several authors as a variant of the seminal result by Johnson and Lindenstrauss (JL) from the mid 1980s. In this work we show how to significantly improve the running time to O(dlog k) for k=O(d 1/2−δ ), for any arbitrary small fixed δ. This beats the better of FJLT and JL. Our analysis uses a powerful measure concentration bound due to Talagrand applied to Rademacher series in Banach spaces (sums of vectors in Banach spaces with random signs). The set of vectors used is a real embedding of dual BCH code vectors over GF(2). We also discuss the number of random bits used and reduction to ℓ 1 space. The connection between geometry and discrete coding theory discussed here is interesting in its own right and may be useful in other algorithmic applications as well.