Fast Cross-Polytope Locality-Sensitive Hashing

We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys 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 O(d2) to 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 O(ln9(d)) random bits