Near Neighbor Search via Efficient Average Distortion Embeddings

TL;DR

This paper introduces a new framework linking near neighbor search efficiency to average distortion embeddings, enabling polynomial pre-processing for a broad class of norms, including $ ext{l}_p$, with concrete improvements for $ ext{l}_p$ spaces.

Contribution

It establishes a reduction from NNS data structures to average distortion embeddings, providing polynomial pre-processing and explicit embeddings for $ ext{l}_p$ spaces with $p \,\geq\, 2$.

Findings

Developed a framework connecting NNS to average distortion embeddings.

Provided explicit, efficiently computable embeddings of $ ext{l}_p$ into $ ext{l}_2$ with quadratic average distortion.

Achieved NNS data structures with polynomial pre-processing for $ ext{l}_p$ norms, matching previous approximation and complexity bounds.

Abstract

A recent series of papers by Andoni, Naor, Nikolov, Razenshteyn, and Waingarten (STOC 2018, FOCS 2018) has given approximate near neighbour search (NNS) data structures for a wide class of distance metrics, including all norms. In particular, these data structures achieve approximation on the order of $p$ for $\ell_p^d$ norms with space complexity nearly linear in the dataset size $n$ and polynomial in the dimension $d$, and query time sub-linear in $n$ and polynomial in $d$. The main shortcoming is the exponential in $d$ pre-processing time required for their construction. In this paper, we describe a more direct framework for constructing NNS data structures for general norms. More specifically, we show via an algorithmic reduction that an efficient NNS data structure for a given metric is implied by an efficient average distortion embedding of it into $\ell_1$ or into Euclidean space. In particular, the resulting data structures require only polynomial pre-processing time, as long as the embedding can be computed in polynomial time. As a concrete instantiation of this framework, we give an NNS data structure for $\ell_p$ with efficient pre-processing that matches the approximation factor, space and query complexity of the aforementioned data structure of Andoni et al. On the way, we resolve a question of Naor (Analysis and Geometry in Metric Spaces, 2014) and provide an explicit, efficiently computable embedding of $\ell_p$, for $p \ge 2$, into $\ell_2$ with (quadratic) average distortion on the order of $p$. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.