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Fast Cross-Polytope Locality-Sensitive Hashing

Authors Christopher Kennedy, Rachel Ward

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Christopher Kennedy
Rachel Ward

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Christopher Kennedy and Rachel Ward. Fast Cross-Polytope Locality-Sensitive Hashing. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ITCS.2017.53

Abstract

We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys \mathcal{O}(d \ln d ) hash computation time.  Building on a recent result in (Andoni, Indyk, Laarhoven, Razenshteyn '15), we show that optimal asymptotic sensitivity for cross-polytope LSH is retained even when the dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform followed by discrete pseudo-rotation, reducing the hash computation time from \mathcal{O}(d^2) to \mathcal{O}(d \ln d ).  Moreover, our scheme achieves the optimal rate of convergence for sensitivity. By incorporating a low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to require only \mathcal{O}(\ln^9(d)) random bits.

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Keywords
  • Locality-sensitive hashing
  • Dimension reduction
  • Johnson-Lindenstrauss lemma

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References

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