Abstract
Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC'98) to give the first sublinear time algorithm for the capproximate 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/c^2 for the l_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 24dimensional 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/c^2 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.
BibTeX  Entry
@InProceedings{chandrasekaran_et_al:LIPIcs:2018:8347,
author = {Karthekeyan Chandrasekaran and Daniel Dadush and Venkata Gandikota and Elena Grigorescu},
title = {{Latticebased Locality Sensitive Hashing is Optimal}},
booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
pages = {42:142:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770606},
ISSN = {18688969},
year = {2018},
volume = {94},
editor = {Anna R. Karlin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8347},
URN = {urn:nbn:de:0030drops83470},
doi = {10.4230/LIPIcs.ITCS.2018.42},
annote = {Keywords: Locality Sensitive Hashing, Approximate Nearest Neighbor Search, Random Lattices}
}
Keywords: 

Locality Sensitive Hashing, Approximate Nearest Neighbor Search, Random Lattices 
Seminar: 

9th Innovations in Theoretical Computer Science Conference (ITCS 2018) 
Issue Date: 

2018 
Date of publication: 

05.01.2018 