The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Randomized algorithms
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Modern computer algebra
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
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Tabulation based 4-universal hashing with applications to second moment estimation
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Finding frequent items in data streams
Theoretical Computer Science - Special issue on automata, languages and programming
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
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
Feature hashing for large scale multitask learning
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
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
Fast moment estimation in data streams in optimal space
Proceedings of the forty-third annual ACM symposium on Theory of computing
Almost optimal explicit Johnson-Lindenstrauss families
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
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
Sketching via hashing: from heavy hitters to compressed sensing to sparse fourier transform
Proceedings of the 32nd symposium on Principles of database systems
Simple and deterministic matrix sketching
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
Low rank approximation and regression in input sparsity time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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We give two different Johnson-Lindenstrauss distributions, each with column sparsity s = Θ(ε−1 log(1/δ)) and embedding into optimal dimension k = O(ε−2 log(1/δ)) to achieve distortion 1±ε with probability 1−δ. That is, only an O(ε)-fraction of entries are non-zero in each embedding matrix in the supports of our distributions. These are the first distributions to provide o(k) sparsity for all values of ε,δ. Previously the best known construction obtained s = Θ(ε-1 log2 (1/δ))1 [Dasgupta-Kumar-Sarlós, STOC 2010]2. In addition, one of our distributions can be sampled from a seed of O(log(1/δ) log d) uniform random bits. Some applications that use Johnson-Lindenstrauss embeddings as a black box, such as those in approximate numerical linear algebra ([Sarlós, FOCS 2006], [Clarkson-Woodruff, STOC 2009]), require exponentially small δ. Our linear dependence on log(1/δ) in the sparsity is thus crucial in these applications to obtain speedup.