## 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 log^{2}(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 language | English |
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Title of host publication | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |

Editors | K.G. Larsen, H.L. Bodlaender, J.F. Raskin |

Place of Publication | Dagstuhl |

Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Pages | 7:1-7:20 |

Number of pages | 20 |

ISBN (Electronic) | 978-3-95977-046-0 |

DOIs | |

Publication status | Published - 1 Nov 2017 |

Externally published | Yes |

Event | 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) - Aalborg, Denmark Duration: 21 Aug 2017 → 25 Aug 2017 Conference number: 42 http://mfcs2017.cs.aau.dk/ |

### Publication series

Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Volume | 83 |

ISSN (Electronic) | 1868-8969 |

### Conference

Conference | 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) |
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Abbreviated title | MFCS 2017 |

Country/Territory | Denmark |

City | Aalborg |

Period | 21/08/17 → 25/08/17 |

Internet address |

## Keywords

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