Approximate Nearest Neighbors in Limited Space

TL;DR

This paper introduces a space-efficient data structure for approximate nearest neighbor search in high-dimensional integer spaces, significantly reducing memory usage while maintaining accuracy.

Contribution

It presents a novel data structure that uses substantially less space than previous methods for approximate nearest neighbor search in bounded integer spaces.

Findings

Achieves $O( ext{epsilon}^{-2} n ext{log}(n) ext{log}(1/ ext{epsilon}))$ bits of space

Improves upon the previous space bound of $O( ext{epsilon}^{-2} n ext{log}(n)^2)$

Provides bounds for the problem of estimating all distances from query points to data points

Abstract

We consider the $(1+\epsilon)$-approximate nearest neighbor search problem: given a set $X$ of $n$ points in a $d$-dimensional space, build a data structure that, given any query point $y$, finds a point $x \in X$ whose distance to $y$ is at most $(1+\epsilon) \min_{x \in X} \|x-y\|$ for an accuracy parameter $\epsilon \in (0,1)$. Our main result is a data structure that occupies only $O(\epsilon^{-2} n \log(n) \log(1/\epsilon))$ bits of space, assuming all point coordinates are integers in the range $\{-n^{O(1)} \ldots n^{O(1)}\}$, i.e., the coordinates have $O(\log n)$ bits of precision. This improves over the best previously known space bound of $O(\epsilon^{-2} n \log(n)^2)$, obtained via the randomized dimensionality reduction method of Johnson and Lindenstrauss (1984). We also consider the more general problem of estimating all distances from a collection of query points to all data points $X$, and provide almost tight upper and lower bounds for the space complexity of this problem.