Lattice-based Locality Sensitive Hashing is Optimal

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• Daniel Dadush, CWI, The Netherlands
• Tuesday 30 January 2018, 14:00-15:00
• Seminar Room 909b, Level 9, EEE Building.

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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/c² for the l_2 norm, which was later shown to be optimal. 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.

Daniel Dadush is currently a tenure track researcher at CWI in the Networks & Optimization group. Previously, he was a Simons Postdoctoral Fellow for 2 years at the Courant Institute of Mathematical Sciences at New York University. He completed his PhD in 2012 within the ACO (Algorithms, Combinatorics, and Optimization) program at Georgia Tech under Santosh Vempala (Professor of Computer Science).

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