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# Lattice-Based Locality Sensitive Hashing

03 Apr
Wednesday, 04/03/2019 12:15pm to 1:15pm
Computer Science Building Room 140
Theory Seminar
Speaker: Venkata Gandikota

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 partitioning of space that achieves an LSH exponent of 1/c^2  for the \ell-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.

Based on a joint work with Karthekeyan Chandrasekaran, Daniel Dadush and Elena Grigorescu.