Hypercube LSH for approximate near neighbors

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2 Citations (Scopus)

Abstract

A celebrated technique for finding near neighbors for the angular distance involves using a set of random hyperplanes to partition the space into hash regions [Charikar, STOC 2002]. Experiments later showed that using a set of orthogonal hyperplanes, thereby partitioning the space into the Voronoi regions induced by a hypercube, leads to even better results [Terasawa and Tanaka, WADS 2007]. However, no theoretical explanation for this improvement was ever given, and it remained unclear how the resulting hypercube hash method scales in high dimensions. In this work, we provide explicit asymptotics for the collision probabilities when using hypercubes to partition the space. For instance, two near-orthogonal vectors are expected to collide with probability ( 1/n)d+o(d) in dimension d, compared to (12)d when using random hyperplanes. Vectors at angle Φ3 collide with probability (√3)d+o(d), compared to (23 )d for random hyperplanes, and near-parallel vectors collide with similar asymptotic probabilities in both cases. For c-approximate nearest neighbor searching, this translates to a decrease in the exponent → of locality-sensitive hashing (LSH) methods of a factor up to log2(φ) ~ 1.652 compared to hyperplane LSH. For c = 2, we obtain → p = 0.302 + o(1) for hypercube LSH, improving upon the p = 0.377 for hyperplane LSH. We further describe how to use hypercube LSH in practice, and we consider an example application in the area of lattice algorithms.

Original languageEnglish
Title of host publication42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017
EditorsK.G. Larsen, H.L. Bodlaender, J.F. Raskin
Place of PublicationDagstuhl
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Pages7:1-7:20
Number of pages20
ISBN (Electronic)978-3-95977-046-0
DOIs
Publication statusPublished - 1 Nov 2017
Externally publishedYes
Event42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) - Aalborg, Denmark
Duration: 21 Aug 201725 Aug 2017
Conference number: 42
http://mfcs2017.cs.aau.dk/

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
Volume83
ISSN (Electronic)1868-8969

Conference

Conference42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)
Abbreviated titleMFCS 2017
CountryDenmark
CityAalborg
Period21/08/1725/08/17
Internet address

Keywords

  • (Approximate) near neighbors
  • Dimensionality reduction
  • Large deviations
  • Lattice algorithms
  • Locality-sensitive hashing

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