The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
Journal of Computer and System Sciences - Special issu on PODS 2001
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
On variants of the Johnson–Lindenstrauss lemma
Random Structures & Algorithms
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
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
Communications of the ACM
Fast Dimension Reduction Using Rademacher Series on Dual BCH Codes
Discrete & Computational Geometry
A sparse Johnson: Lindenstrauss transform
Proceedings of the forty-second ACM symposium on Theory of computing
Johnson-Lindenstrauss lemma for circulant matrices
Random Structures & 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
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Sparser Johnson-Lindenstrauss Transforms
Journal of the ACM (JACM)
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The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from probability in Banach spaces that were successfully used in the context of sparse reconstruction to advance on an open problem in random pojection. In particular, we generalize and use an intricate result by Rudelson and Veshynin [2008] for sparse reconstruction which uses Dudley’s theorem for bounding Gaussian processes. Our main result states that any set of N = exp(Õ(n)) real vectors in n dimensional space can be linearly mapped to a space of dimension k = O(log N polylog(n)), while (1) preserving the pairwise distances among the vectors to within any constant distortion and (2) being able to apply the transformation in time O(n log n) on each vector. This improves on the best known bound N = exp(Õ(n1/2)) achieved by Ailon and Liberty [2009] and N = exp(Õ(n1/3)) by Ailon and Chazelle [2010]. The dependence in the distortion constant however is suboptimal, and since the publication of an early version of the work, the gap between upper and lower bounds has been considerably tightened obtained by Krahmer and Ward [2011]. For constant distortion, this settles the open question posed by these authors up to a polylog(n) factor while considerably simplifying their constructions.