Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors

Abstract

We show tight upper and lower bounds for time-space trade-offs for the c-approximate Near Neighbor Search problem. For the d-dimensional Euclidean space and npoint datasets, we develop a data structure with space n 1+ρu+o(1) + O(dn) and query time n ρq+o(1) + dn o(1) for every ρ u , ρ q ≥ 0 with:In particular, for the approximation c = 2 we get:• Space n 1.77... and query time n o(1) , significantly improving upon known data structures that support very fast queries [IM98, KOR00];• Space n 1.14... and query time n 0.14... , matching the optimal data-dependent Locality-Sensitive Hashing (LSH) from [AR15];• Space n 1+o(1) and query time n 0.43... , making significant progress in the regime of near-linear space, which is arguably of the most interest for prac-This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor c > 1, improving upon [Kap15]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [BDGL16] and datadependent hashing [AINR14, AR15]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole trade-off (0.1) in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower * This paper merges two arXiv preprints: [Laa15c] (appeared online on November 24, 2015) and [ALRW16] (appeared online on May 9, 2016), and subsumes both of these articles. The full version containing all the proofs is available at https://arxiv.org/abs/1608.03580 bounds for one and two probes that match (0.1) for ρ q = 0, improving upon the best known lower bounds from [PTW10]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.

Hashing-Based-Estimators for Kernel Density in High Dimensions

“…We show that the latter is at least as hard as the (r, c)-ANNS with n 1 µ points and c O( log(n) log(1/ε) ) under the Hamming distance. Combined with the results of Panigrahy, Talwar, Wieder [32] and Andoni et al [5], we get non-trivial lower bounds in the cell-probe model with a single probe that captures an interesting class of algorithms based on adaptive coresets.…”

Explaining the Success of Nearest Neighbor Methods in Prediction

“…Depending on the feature space and distance function chosen or learned by the practitioner, different fast approximate nearest neighbor search algorithms are available. These search algorithms, both for general high-dimensional feature spaces (e.g., Gionis et al 1999;Datar et al 2004;Bawa et al 2005;Andoni and Indyk 2008;Ailon and Chazelle 2009;Muja and Lowe 2009;Boytsov and Naidan 2013;Dasgupta and Sinha 2015;Mathy et al 2015;Andoni et al 2017) and specialized to image patches (e.g., Barnes et al 2009;Ta et al 2014), can rapidly determine which data points are close to each other while parallelizing across search queries. These methods often use locality-sensitive hashing (Indyk and Motwani, 1998), which comes with a theoretical guarantee on approximation accuracy, or randomized trees (e.g., Bawa et al 2005;Muja and Lowe 2009;Dasgupta and Sinha 2015;Mathy et al 2015), which quickly prune search spaces when the trees are sufficiently balanced.…”

Non-Parametric Inference Adaptive to Intrinsic Dimension

“…One such application is given in high-dimensional settings where the exact nearest neighbor search is computationally expensive and Approximate Nearest Neighbor (ANN) search is often replaced in order to reduce this cost. Our flexible result allows us to use the state-of-the-art ANN algorithms (see e.g., Andoni et al [2017Andoni et al [ , 2018) while maintaining consistency and asymptotic normality.…”