Approximate nearest neighbors search without false negatives for $l_2$ for $c>\sqrt{\log\log{n}}$

TL;DR

This paper introduces new data structures for approximate nearest neighbor search in high-dimensional Euclidean space that operate without false negatives for larger approximation factors, improving previous algorithms.

Contribution

It presents novel data structures for false-negative-free approximate nearest neighbor search in Euclidean space applicable for larger c values, with efficient query and preprocessing times.

Findings

Works for c = ω(√log log n)

Poly-logarithmic query time

Polynomial preprocessing time

Abstract

In this paper, we report progress on answering the open problem presented by Pagh~[14], who considered the nearest neighbor search without false negatives for the Hamming distance. We show new data structures for solving the $c$-approximate nearest neighbors problem without false negatives for Euclidean high dimensional space $\mathcal{R}^d$. These data structures work for any $c = \omega(\sqrt{\log{\log{n}}})$, where $n$ is the number of points in the input set, with poly-logarithmic query time and polynomial preprocessing time. This improves over the known algorithms, which require $c$ to be $\Omega(\sqrt{d})$. This improvement is obtained by applying a sequence of reductions, which are interesting on their own. First, we reduce the problem to $d$ instances of dimension logarithmic in $n$. Next, these instances are reduced to a number of $c$-approximate nearest neighbor search instances in $\big(\mathbb{R}^k\big)^L$ space equipped with metric $m(x,y) = \max_{1 \le i \le L}(\lVert x_i - y_i\rVert_2)$.