Practical and Optimal LSH for Angular Distance

TL;DR

This paper introduces a practical and theoretically optimal Locality-Sensitive Hashing family for angular distance, improving near neighbor search efficiency and providing a trade-off analysis with lower bounds.

Contribution

It presents a new LSH family for angular distance that is both practical and asymptotically optimal, along with a multiprobe extension and lower bound analysis.

Findings

New LSH family for angular distance with optimal exponent

Practical implementation outperforming previous methods

Lower bound demonstrating near-optimal trade-offs

Abstract

We show the existence of a Locality-Sensitive Hashing (LSH) family for the angular distance that yields an approximate Near Neighbor Search algorithm with the asymptotically optimal running time exponent. Unlike earlier algorithms with this property (e.g., Spherical LSH [Andoni, Indyk, Nguyen, Razenshteyn 2014], [Andoni, Razenshteyn 2015]), our algorithm is also practical, improving upon the well-studied hyperplane LSH [Charikar, 2002] in practice. We also introduce a multiprobe version of this algorithm, and conduct experimental evaluation on real and synthetic data sets. We complement the above positive results with a fine-grained lower bound for the quality of any LSH family for angular distance. Our lower bound implies that the above LSH family exhibits a trade-off between evaluation time and quality that is close to optimal for a natural class of LSH functions.