Approximate Near Neighbors for General Symmetric Norms

TL;DR

This paper introduces an efficient data structure for approximate nearest neighbor search in symmetric normed spaces, achieving near-logarithmic approximation with sublinear query time and near-linear space, using novel low-distortion embeddings.

Contribution

It provides the first data structure with doubly-logarithmic approximation for symmetric norms, leveraging low-distortion embeddings into product spaces, and establishes limitations for general norms.

Findings

Achieves $n^{o(1)}$ query time for symmetric norms.

Provides low-distortion embeddings into product spaces.

Shows limitations for extending techniques to general norms.

Abstract

We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation. Specifically, for every $n$, $d = n^{o(1)}$, and every $d$-dimensional symmetric norm $\|\cdot\|$, there exists a data structure for $\mathrm{poly}(\log \log n)$-approximate nearest neighbor search over $\|\cdot\|$ for $n$-point datasets achieving $n^{o(1)}$ query time and $n^{1+o(1)}$ space. The main technical ingredient of the algorithm is a low-distortion embedding of a symmetric norm into a low-dimensional iterated product of top-$k$ norms. We also show that our techniques cannot be extended to general norms.