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 $mathcal{O}(d ln d )$ 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