Fast Cross-polytope Locality-sensitive Hashing

Kennedy Christopher, Ward Rachel. Arxiv 2016

[Paper]    
ARXIV Independent LSH

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 (by Andoni, Indyk, Laarhoven, Razenshteyn, Schmidt, 2015), 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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