Optimal Data-Dependent Hashing for Approximate Near Neighbors

TL;DR

This paper introduces an optimal data-dependent hashing scheme for approximate near neighbor search that outperforms classical LSH methods in Euclidean and Hamming spaces, with improved query time and space complexity.

Contribution

It presents a new data-dependent hashing scheme that is proven to be optimal and surpasses existing LSH-based methods for all approximation factors.

Findings

Achieves optimal query time and space complexity for Euclidean space.

Outperforms classical LSH methods for all approximation factors.

Provides a new technical approach by decomposing datasets into pseudo-random subsets.

Abstract

We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an $n$-point data set in a $d$-dimensional space our data structure achieves query time $O(d n^{\rho+o(1)})$ and space $O(n^{1+\rho+o(1)} + dn)$, where $\rho=\tfrac{1}{2c^2-1}$ for the Euclidean space and approximation $c>1$. For the Hamming space, we obtain an exponent of $\rho=\tfrac{1}{2c-1}$. Our result completes the direction set forth in [AINR14] who gave a proof-of-concept that data-dependent hashing can outperform classical Locality Sensitive Hashing (LSH). In contrast to [AINR14], the new bound is not only optimal, but in fact improves over the best (optimal) LSH data structures [IM98,AI06] for all approximation factors $c>1$. From the technical perspective, we proceed by decomposing an arbitrary dataset into several subsets that are, in a certain sense, pseudo-random.