On Symmetric and Asymmetric LSHs for Inner Product Search

TL;DR

This paper investigates locality sensitive hashing methods for inner product search, demonstrating the existence of a simple symmetric LSH with better guarantees and empirical performance, and clarifying when asymmetric LSHs are necessary.

Contribution

The paper proves the existence of a simple symmetric LSH for inner product similarity with improved guarantees and empirical results, challenging prior assumptions about the necessity of asymmetry.

Findings

A simple symmetric LSH outperforms the asymmetric one proposed by Shrivastava and Li.

Symmetric LSHs can be stronger and more effective than asymmetric variants in certain settings.

Asymmetry is only needed in specific variants of the problem, with different LSH requirements.

Abstract

We consider the problem of designing locality sensitive hashes (LSH) for inner product similarity, and of the power of asymmetric hashes in this context. Shrivastava and Li argue that there is no symmetric LSH for the problem and propose an asymmetric LSH based on different mappings for query and database points. However, we show there does exist a simple symmetric LSH that enjoys stronger guarantees and better empirical performance than the asymmetric LSH they suggest. We also show a variant of the settings where asymmetry is in-fact needed, but there a different asymmetric LSH is required.