Lattice-based locality sensitive hashing is optimal

Karthekeyan Chandrasekaran, Daniel Dadush, Venkata Gandikota, Elena Grigorescu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC ‘98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes “nearby” points to the same bucket and “far away” points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS ‘06) constructed an LSH family based on random ball partitionings of space that achieves an LSH exponent of 1/c2 for the 2 norm, which was later shown to be optimal by Motwani, Naor and Panigrahy (SIDMA ‘07) and O’Donnell, Wu and Zhou (TOCT ‘14). Although optimal in the LSH exponent, the ball partitioning approach is computationally expensive. So, in the same work, Andoni and Indyk proposed a simpler and more practical hashing scheme based on Euclidean lattices and provided computational results using the 24-dimensional Leech lattice. However, no theoretical analysis of the scheme was given, thus leaving open the question of finding the exponent of lattice based LSH. In this work, we resolve this question by showing the existence of lattices achieving the optimal LSH exponent of 1/c2 using techniques from the geometry of numbers. At a more conceptual level, our results show that optimal LSH space partitions can have periodic structure. Understanding the extent to which additional structure can be imposed on these partitions, e.g. to yield low space and query complexity, remains an important open problem.

Original languageEnglish (US)
Title of host publication9th Innovations in Theoretical Computer Science, ITCS 2018
EditorsAnna R. Karlin
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770606
DOIs
StatePublished - Jan 1 2018
Externally publishedYes
Event9th Innovations in Theoretical Computer Science, ITCS 2018 - Cambridge, United States
Duration: Jan 11 2018Jan 14 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume94
ISSN (Print)1868-8969

Conference

Conference9th Innovations in Theoretical Computer Science, ITCS 2018
CountryUnited States
CityCambridge
Period1/11/181/14/18

Keywords

  • Approximate Nearest Neighbor Search
  • Locality Sensitive Hashing
  • Random Lattices

ASJC Scopus subject areas

  • Software

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