Beyond Locality-Sensitive Hashing

TL;DR

This paper introduces a novel data structure for approximate near neighbor search in Euclidean and Hamming spaces, surpassing previous bounds and lower bounds, with improved query time and space complexity.

Contribution

It presents the first data structure to improve upon prior bounds for the c-approximate near neighbor problem, breaking existing locality-sensitive hashing lower bounds.

Findings

Achieves faster query time than previous methods.

Reduces space complexity for high-dimensional data.

Provides improved bounds for Hamming and  spaces.

Abstract

We present a new data structure for the c-approximate near neighbor problem (ANN) in the Euclidean space. For n points in R^d, our algorithm achieves O(n^{\rho} + d log n) query time and O(n^{1 + \rho} + d log n) space, where \rho <= 7/(8c^2) + O(1 / c^3) + o(1). This is the first improvement over the result by Andoni and Indyk (FOCS 2006) and the first data structure that bypasses a locality-sensitive hashing lower bound proved by O'Donnell, Wu and Zhou (ICS 2011). By a standard reduction we obtain a data structure for the Hamming space and \ell_1 norm with \rho <= 7/(8c) + O(1/c^{3/2}) + o(1), which is the first improvement over the result of Indyk and Motwani (STOC 1998).