Locality-Sensitive Hashing without False Negatives for l_p

TL;DR

This paper introduces locality-sensitive hash functions that guarantee no false negatives for l_p norms, enabling efficient approximate nearest neighbor searches with provable guarantees and improved algorithms.

Contribution

It presents the first construction of false-negative-free LSH functions for l_p spaces and applies them to develop efficient, provably correct approximate nearest neighbor algorithms.

Findings

Hash functions ensure collision for all point pairs within radius R.

Algorithms achieve sublinear expected query time with efficient preprocessing.

Progress on open problem for false-negative-free nearest neighbor search in Hamming distance.

Abstract

In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius $R$ in $d$ dimensional space equipped with $l_p$ norm when $p \in [1,\infty]$. Furthermore, we show how to use these hash functions to solve the $c$-approximate nearest neighbor search problem without false negatives. Namely, if there is a point at distance $R$, we will certainly report it and points at distance greater than $cR$ will not be reported for $c=\Omega(\sqrt{d},d^{1-\frac{1}{p}})$. The constructed algorithms work: - with preprocessing time $\mathcal{O}(n \log(n))$ and sublinear expected query time, - with preprocessing time $\mathcal{O}(\mathrm{poly}(n))$ and expected query time $\mathcal{O}(\log(n))$. Our paper reports progress on answering the open problem presented by Pagh [8] who considered the nearest neighbor search without false negatives for the Hamming distance.