Fast Cross-Polytope Locality-Sensitive Hashing

TL;DR

This paper introduces a fast, asymptotically optimal cross-polytope locality-sensitive hashing scheme that significantly reduces computation time using Johnson-Lindenstrauss transforms, maintaining optimal sensitivity.

Contribution

It presents a novel LSH variant that achieves optimal asymptotic sensitivity with reduced computational complexity, leveraging fast Johnson-Lindenstrauss transforms.

Findings

Achieves $ ext{O}(d \, \ln d)$ hash computation time.

Maintains optimal asymptotic sensitivity.

Requires only $ ext{O}(\ln^9(d))$ random bits with low-randomness transforms.

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 (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