摘要：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.
