A GPGPU Algorithm for c-Approximate r-Nearest Neighbor Search in High Dimensions

Carraher, Lee A.; Wilsey, Philip A.; Annexstein, Fred S.

doi:10.1109/ipdpsw.2013.239

Cited by 2 publications

(1 citation statement)

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“…To evaluate lattice based LSH, Andoni and Indyk [4] provided experimental results for L being the 24 dimensional Leech lattice equipped with the decoder of [3]. A version of this scheme with the 8 dimensional E8 lattice has also been implemented and tested in [21], and a parallelized GPU implementation of the Leech lattice scheme was tested in [11].…”

Section: Introductionmentioning

confidence: 99%

Lattice-based Locality Sensitive Hashing is Optimal

Chandrasekaran¹,

Dadush²,

Gandikota³

et al. 2017

Preprint

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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/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 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/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.

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