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Near Neighbor Search via Efficient Average Distortion Embeddings

Authors Deepanshu Kush, Aleksandar Nikolov, Haohua Tang

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  • Theory of computation → Nearest neighbor algorithms
Keywords
  • Nearest neighbor search
  • metric space embeddings
  • average distortion embeddings
  • locality-sensitive hashing

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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 𝓁_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 metric ℳ is implied by an efficient average distortion embedding of ℳ into 𝓁₁ or the 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 𝓁_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 𝓁_p, for p ≥ 1, into 𝓁₁ with average distortion on the order of p. Furthermore, we also give data structures for Schatten-p spaces with improved space and query complexity, albeit still requiring exponential pre-processing when p ≥ 2. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.

Cite As Get BibTex

Deepanshu Kush, Aleksandar Nikolov, and Haohua Tang. Near Neighbor Search via Efficient Average Distortion Embeddings. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.50

Author Details

Deepanshu Kush
  • University of Toronto, Canada
Aleksandar Nikolov
  • University of Toronto, Canada
Haohua Tang
  • University of Toronto, Canada

Funding

  • Nikolov, Aleksandar: Supported by an NSERC Discovery Grant (RGPIN-2016-06333), and a Tier II Canada Research Chair.
  • Tang, Haohua: Supported by a Tier II Canada Research Chair.

Acknowledgements

We want to thank Sushant Sachdeva for a fruitful discussion.

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