The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces
SIAM Journal on Computing
Fast Monte-Carlo Algorithms for finding low-rank approximations
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
On Approximate Nearest Neighbors in Non-Euclidean Spaces
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
An Algorithmic Theory of Learning: Robust Concepts and Random Projection
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Fast Monte-Carlo Algorithms for Approximate Matrix Multiplication
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Hardness of Approximating the Shortest Vector Problem in Lattices
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Lower bounds for linear degeneracy testing
Journal of the ACM (JACM)
Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Improved Approximation Algorithms for Large Matrices via Random Projections
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Uncertainty principles, extractors, and explicit embeddings of l2 into l1
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Dense Fast Random Projections and Lean Walsh Transforms
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Near Optimal Dimensionality Reductions That Preserve Volumes
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
The Mailman algorithm: A note on matrix--vector multiplication
Information Processing Letters
A fast and efficient algorithm for low-rank approximation of a matrix
Proceedings of the forty-first annual ACM symposium on Theory of computing
Fast Algorithms for Approximating the Singular Value Decomposition
ACM Transactions on Knowledge Discovery from Data (TKDD)
Blendenpik: Supercharging LAPACK's Least-Squares Solver
SIAM Journal on Scientific Computing
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
SETA'12 Proceedings of the 7th international conference on Sequences and Their Applications
Fast approximation of matrix coherence and statistical leverage
The Journal of Machine Learning Research
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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 ℓd2 to ℓd2 in time O(max{d log d,k3}). For k in [Ω(log d), O(d1/2)] 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 80's. In this work we show how to significantly improve the running time to O(d log k) for k = O(d1/2−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 ℓ1 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.