Communications of the ACM
A sparse Johnson: Lindenstrauss transform
Proceedings of the forty-second ACM symposium on Theory of computing
Stochastic algorithms in linear algebra: beyond the Markov chains and von Neumann-Ulam scheme
NMA'10 Proceedings of the 7th international conference on Numerical methods and applications
Subspace embeddings for the L1-norm with applications
Proceedings of the forty-third annual ACM symposium on Theory of computing
Fast locality-sensitive hashing
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
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
Acceleration of randomized Kaczmarz method via the Johnson---Lindenstrauss Lemma
Numerical 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
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
Explicit Dimension Reduction and Its Applications
SIAM Journal on Computing
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
Fast approximation of matrix coherence and statistical leverage
The Journal of Machine Learning Research
Sparser Johnson-Lindenstrauss Transforms
Journal of the ACM (JACM)
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We introduce a new low-distortion embedding of $\ell_2^d$ into $\ell_p^{O(\log n)}$ ($p=1,2$) called the fast Johnson-Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the “Heisenberg principle” of the Fourier transform, i.e., its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in $\ell_1$ and $\ell_2$. We consider the case of approximate nearest neighbors in $\ell_2^d$. We provide a faster algorithm using classical projections, which we then speed up further by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.