Abstract
Locality-sensitive hashing (LSH), introduced by Indyk and Motwani in STOC ’98, has been an extremely influential framework for nearest neighbor search in high-dimensional data sets. While theoretical work has focused on the approximate nearest neighbor problem, in practice LSH data structures with suitably chosen parameters are used to solve the exact nearest neighbor problem (with some error probability). Sublinear query time is often possible in practice even for exact nearest neighbor search, intuitively because the nearest neighbor tends to be significantly closer than other data points. However, theory offers little advice on how to choose LSH parameters outside of pre-specified worst-case settings.
We introduce the technique of confirmation sampling for solving the exact nearest neighbor problem using LSH. First, we give a general reduction that transforms a sequence of data structures that each find the nearest neighbor with a small, unknown probability, into a data structure that returns the nearest neighbor with probability \(1-\delta \), using as few queries as possible. Second, we present a new query algorithm for the LSH Forest data structure with L trees that is able to return the exact nearest neighbor of a query point within the same time bound as an LSH Forest of \(\varOmega (L)\) trees with internal parameters specifically tuned to the query and data.
T. Christiani—The research leading to these results has received funding from the European Research Council under the European Union’s 7th Framework Programme (FP7/2007-2013)/ERC grant agreement no. 614331.
R. Pagh—Supported by Villum Foundation grant 16582 to Basic Algorithms Research Copenhagen (BARC). Part of this work was done while visiting Simons Institute for the Theory of Computing.
M. Thorup—Supported by an Investigator Grant from the Villum Foundation, Grant No. 16582.
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Notes
- 1.
The sampling of a random element ensures compatibility with ConfirmationSampling, which requires a sample to be returned even if there is no hash collision. It is not really necessary from an algorithmic viewpoint, but also does not hurt the asymptotic performance.
- 2.
For every choice of constant \(c \ge 1\) there exists a constant \(n_0\) such that for \(n \ge n_0\) we can obtain success probability \(1 - 1/n^c\) where \(n = |P|\) denotes the size of the set of data points.
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Christiani, T., Pagh, R., Thorup, M. (2020). Confirmation Sampling for Exact Nearest Neighbor Search. In: Satoh, S., et al. Similarity Search and Applications. SISAP 2020. Lecture Notes in Computer Science(), vol 12440. Springer, Cham. https://doi.org/10.1007/978-3-030-60936-8_8
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