Authors
Nir Ailon, Edo Liberty
Publication date
2009/12/1
Journal
Discrete & Computational Geometry
Volume
42
Issue
4
Pages
615-630
Publisher
Springer New York
Description
The Fast Johnson–Lindenstrauss Transform (FJLT) was recently discovered by Ailon and Chazelle as a novel technique for performing fast dimension reduction with small distortion from 2 d to 2 k in time O(max {dlog d,k 3}). For k in [Ω(log d),O(d 1/2)], this beats time O(dk) achieved by naive multiplication by random dense matrices, an approach followed by several authors as a variant of the seminal result by Johnson and Lindenstrauss (JL) from the mid 1980s. In this work we show how to significantly improve the running time to O(dlog k) for k=O(d 1/2−δ ), for any arbitrary small fixed δ. This beats the better of FJLT and JL. Our analysis uses a powerful measure concentration bound due to Talagrand applied to …
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