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Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform
Abstract
We introduce a new low-distortion embedding of l2d into lpO(log n) (p=1,2), called the Fast-Johnson-Linden-strauss-Transform. 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, ie, its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in l1 and l2. We consider the case of approximate nearest neighbors in l2d. We provide a faster algorithm using classical projections, which we then further speed up by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.
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Cites
Monte Carlo sampling methods using Markov chains and their applications.
Hastings, W. K. (1970) · Biometrika · doi:10.1093/biomet/57.1.97 · Cited by 15,195
Approximate nearest neighbors: towards removing the curse of dimensionality
Indyk, P., & Motwani, R. (1998) · Proceedings of the thirtieth annual ACM symposium on Theory of computing - STOC '98 · doi:10.1145/276698.276876 · Cited by 4,188
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Andoni, A., & Indyk, P. (2008) · Communications of the ACM · doi:10.1145/1327452.1327494 · Cited by 1,437
Near-Optimal Hashing Algorithms for Approximate Nearest Neighbor in High Dimensions
Andoni, A., & Indyk, P. (2006) · 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06) · doi:10.1109/focs.2006.49 · Cited by 1,067
Relative-Error $CUR$ Matrix Decompositions
Drineas, P., Mahoney, M. W., & Muthukrishnan, S. (2008) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/07070471x · Cited by 461
Compressive Sensing by Random Convolution
Romberg, J. (2009) · SIAM Journal on Imaging Sciences · doi:10.1137/08072975x · Cited by 458
New and Improved Johnson–Lindenstrauss Embeddings via the Restricted Isometry Property
Krahmer, F., & Ward, R. (2011) · SIAM Journal on Mathematical Analysis · doi:10.1137/100810447 · Cited by 290
Metadata
- Authors
- Ailon, N., & Chazelle, B.
- Venue
- Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
- Publisher
- ACM
- Published
- 2006-05-21
- Pages
- 557-563
- OpenAlex
- W2152402969
- Cited by
- 379
- References
- 2
- FWCI
- 15.37
- Type
- conference paper
- Language
- en
- Open access
- closed
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- Ailon, N., & Chazelle, B. (2006). Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing. https://doi.org/10.1145/1132516.1132597
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