Based on the Kac random walk on the orthogonal group, we present a fast Johnson-Lindenstrauss transform: given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead. In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle}  ] ] .
In this work, we analyze dimension reduction algorithms based on the Kac ^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% ^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } walk and discrete variants. ( 1 for all x\in X, and such that for each x\in X, \Psi x can be computed } ) For n points in \mathbb{R SODA, 2008], and Bamberger and Krahmer arXiv, 2017]. Our algorithm is memory-optimal, and outperforms existing algorithms in regimes when n is sufficiently large and the distortion parameter is sufficiently small. In particular, this confirms a conjecture of Ailon and Chazelle STOC, 2006] in a stronger form. (2) The same construction gives a simple transform with optimal Restricted Isometry Property (RIP) which can be applied in time O(d\log{d} , \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead. In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle} ) for essentially the same range of sparsity as in the best-known such transform due to Ailon and Rauhut Discrete Comput. Geom., 2014]. (3) We show that by fixing the angle in the Kac walk to be \pi/4 throughout, one obtains optimal JL and RIP transforms with almost the same running time, thereby confirming -- up to a \log\log{d SIAM J. Sci. Comput., 2010]. Our moment-based analysis of this modification of the Kac walk may also be of independent interest .
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Based
In this work, we analyze dimension reduction algorithms based
D
random walk on the orthogonal group, we present a fast Johnson-Lindenstrauss transform: given a set X of n points in \mathbb{R
D
\mathbb{R
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walk and discrete variants. (
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+\epsilon)\cdot \|x\|_{2
A
) For n points in \mathbb{R
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SODA, 2008
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, and Bamberger and Krahmer
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arXiv, 2017
A
. Our algorithm is memory-optimal, and outperforms existing algorithms in regimes when n is sufficiently large and the distortion parameter is sufficiently small. In particular, this confirms a conjecture of Ailon and Chazelle
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STOC, 2006
A
in a stronger form. (2) The same construction gives a simple transform with optimal Restricted Isometry Property (RIP) which can be applied
D
+ \min\{d\log{n
A
) for essentially the same range of sparsity as in the best-known such transform due to Ailon and Rauhut
A
Discrete Comput. Geom., 2014
A
. (3) We show that by fixing the angle in the Kac walk to be \pi/4 throughout, one obtains optimal JL and RIP transforms with almost the same running time, thereby confirming -- up to a \log\log{d
A
SIAM J. Sci. Comput., 2010
A
. Our moment-based analysis of this modification of the Kac walk may also be of independent interest