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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.50
URN: urn:nbn:de:0030-drops-138490
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13849/
Kush, Deepanshu ;
Nikolov, Aleksandar ;
Tang, Haohua
Near Neighbor Search via Efficient Average Distortion Embeddings
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 ?_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 ?₁ 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 ?_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 ?_p, for p ≥ 1, into ?₁ 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.
BibTeX - Entry
@InProceedings{kush_et_al:LIPIcs.SoCG.2021.50,
author = {Kush, Deepanshu and Nikolov, Aleksandar and Tang, Haohua},
title = {{Near Neighbor Search via Efficient Average Distortion Embeddings}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {50:1--50:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-184-9},
ISSN = {1868-8969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13849},
URN = {urn:nbn:de:0030-drops-138490},
doi = {10.4230/LIPIcs.SoCG.2021.50},
annote = {Keywords: Nearest neighbor search, metric space embeddings, average distortion embeddings, locality-sensitive hashing}
}
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Keywords: |
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Nearest neighbor search, metric space embeddings, average distortion embeddings, locality-sensitive hashing |
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Collection: |
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37th International Symposium on Computational Geometry (SoCG 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.06.2021 |
