Fast Distance Oracles for Any Symmetric Norm

TL;DR

This paper introduces a fast $(1+\varepsilon)$-approximate distance oracle for any symmetric norm, significantly improving efficiency for similarity search in high-dimensional spaces across various norms.

Contribution

It presents a novel data structure that provides efficient approximate distance queries for all symmetric norms, extending beyond $\ell_p$ norms to many practical norms.

Findings

Preprocessing time and space are $\tilde{O}(n(d + \mathrm{mmc}(l)^2))$.

Query time is $\tilde{O}(d + |S| \cdot \mathrm{mmc}(l)^2)$.

Matches state-of-the-art for $\ell_p$ norms when $l=\ell_p$.

Abstract

In the Distance Oracle problem, the goal is to preprocess $n$ vectors $x_1, x_2, \cdots, x_n$ in a $d$-dimensional metric space $(\mathbb{X}^d, \| \cdot \|_l)$ into a cheap data structure, so that given a query vector $q \in \mathbb{X}^d$ and a subset $S\subseteq [n]$ of the input data points, all distances $\| q - x_i \|_l$ for $x_i\in S$ can be quickly approximated (faster than the trivial $\sim d|S|$ query time). This primitive is a basic subroutine in machine learning, data mining and similarity search applications. In the case of $\ell_p$ norms, the problem is well understood, and optimal data structures are known for most values of $p$. Our main contribution is a fast $(1+\varepsilon)$ distance oracle for any symmetric norm $\|\cdot\|_l$. This class includes $\ell_p$ norms and Orlicz norms as special cases, as well as other norms used in practice, e.g. top-$k$ norms, max-mixture and sum-mixture of $\ell_p$ norms, small-support norms and the box-norm. We propose a novel data structure with $\tilde{O}(n (d + \mathrm{mmc}(l)^2 ) )$ preprocessing time and space, and $t_q = \tilde{O}(d + |S| \cdot \mathrm{mmc}(l)^2)$ query time, for computing distances to a subset $S$ of data points, where $\mathrm{mmc}(l)$ is a complexity-measure (concentration modulus) of the symmetric norm. When $l = \ell_{p}$ , this runtime matches the aforementioned state-of-art oracles.