Smooth Tradeoffs between Insert and Query Complexity in Nearest Neighbor Search

Abstract

Locality Sensitive Hashing (LSH) has emerged as the method of choice for high dimensional similarity search, a classical problem of interest in numerous applications. LSH-based solutions require that each data point be inserted into a number A of hash tables, after which a query can be answered by performing B lookups. The original LSH solution of [IM98] showed for the first time that both A and B can be made sublinear in the number of data points. Unfortunately, the classical LSH solution does not provide any tradeoff between insert and query complexity, whereas for data (respectively, query) intensive applications one would like to minimize insert time by choosing a smaller A (respectively, minimize query time by choosing a smaller B). A partial remedy for this is provided by Entropy LSH [Pan06], which allows to make either inserts or queries essentially constant time at the expense of a loss in the other parameter, but no algorithm that achieves a smooth tradeoff is known.In this paper, we present an algorithm for performing similarity search under the Euclidean metric that resolves the problem above. Our solution is inspired by Entropy LSH, but uses a very different analysis to achieve a smooth tradeoff between insert and query complexity. Our results improve upon or match, up to lower order terms in the exponent, best known data-oblivious algorithms for the Euclidean metric.

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

“…In particular, (1.3) is better than all the prior work on time-space trade-offs for ANN, including the most recent tradeoff [Kap15]. Moreover, using a reduction from [Val15], we achieve the bound (1.3) for the whole R d as opposed to just the unit sphere.…”

“…This regime has been studied since [Ind01a] Despite significant progress in the near-linear space regime, no known algorithms obtain near-linear space and a sublinear query time for all approximations c > 1. For example, the best currently known algorithm of [Kap15] obtained query time of roughly n 4/(c 2 +1) , which becomes trivial for c < √ 3.…”

“…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].…”

“…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].…”

Set Similarity Search for Skewed Data

“…In contrast, strong theoretical results are known when randomization and approximation of distances is allowed, e.g. [2,18,22,25,46]. Even though existing randomized algorithms for set similarity search are superior to commonly used heuristics for difficult data distributions with small skew, it is clear that the heuristics will work much better (in theory and in practice) when the skew is large enough.…”