License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.57
URN: urn:nbn:de:0030-drops-87700
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8770/
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Laarhoven, Thijs

Graph-Based Time-Space Trade-Offs for Approximate Near Neighbors

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LIPIcs-SoCG-2018-57.pdf (0.6 MB)

Abstract

We take a first step towards a rigorous asymptotic analysis of graph-based methods for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of randomized greedy walks on the approximate nearest neighbor graph. For random data sets of size n = 2^{o(d)} on the d-dimensional Euclidean unit sphere, using near neighbor graphs we can provably solve the approximate nearest neighbor problem with approximation factor c > 1 in query time n^{rho_{q} + o(1)} and space n^{1 + rho_{s} + o(1)}, for arbitrary rho_{q}, rho_{s} >= 0 satisfying (2c^2 - 1) rho_{q} + 2 c^2 (c^2 - 1) sqrt{rho_{s} (1 - rho_{s})} >= c^4. Graph-based near neighbor searching is especially competitive with hash-based methods for small c and near-linear memory, and in this regime the asymptotic scaling of a greedy graph-based search matches optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [Andoni et al., 2017]. We further study how the trade-offs scale when the data set is of size n = 2^{Theta(d)}, and analyze asymptotic complexities when applying these results to lattice sieving.

BibTeX - Entry

@InProceedings{laarhoven:LIPIcs:2018:8770,
  author =	{Thijs Laarhoven},
  title =	{{Graph-Based Time-Space Trade-Offs for Approximate Near Neighbors}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{57:1--57:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8770},
  URN =		{urn:nbn:de:0030-drops-87700},
  doi =		{10.4230/LIPIcs.SoCG.2018.57},
  annote =	{Keywords: approximate nearest neighbor problem, near neighbor graphs, locality-sensitive hashing, locality-sensitive filters, similarity search}
}

Keywords: approximate nearest neighbor problem, near neighbor graphs, locality-sensitive hashing, locality-sensitive filters, similarity search
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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