### Abstract

We introduce a new low-distortion embedding of l _{2} ^{d} l _{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 l _{1} and l _{2}. We consider the case of approximate nearest neighbors in l _{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.

Original language | English (US) |
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Pages (from-to) | 302-322 |

Number of pages | 21 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - Jul 15 2009 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

### Keywords

- Approximate nearest neighbors
- Dimension reduction
- Random matrices

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## Cite this

*SIAM Journal on Computing*,

*39*(1), 302-322. https://doi.org/10.1137/060673096