A Framework for Similarity Search with Space-Time Tradeoffs using Locality-Sensitive Filtering

Abstract

We present a framework for similarity search based on Locality-Sensitive Filtering (LSF), generalizing the Indyk-Motwani (STOC 1998) Locality-Sensitive Hashing (LSH) framework to support space-time tradeoffs. Given a family of filters, defined as a distribution over pairs of subsets of space with certain locality-sensitivity properties, we can solve the approximate near neighbor problem in $d$-dimensional space for an $n$-point data set with query time $dn^{\rho_q+o(1)}$, update time $dn^{\rho_u+o(1)}$, and space usage $dn + n^{1 + \rho_u + o(1)}$. The space-time tradeoff is tied to the tradeoff between query time and update time, controlled by the exponents $\rho_q, \rho_u$ that are determined by the filter family. Locality-sensitive filtering was introduced by Becker et al. (SODA 2016) together with a framework yielding a single, balanced, tradeoff between query time and space, further relying on the assumption of an efficient oracle for the filter evaluation algorithm. We extend the LSF framework to support space-time tradeoffs and through a combination of existing techniques we remove the oracle assumption. Building on a filter family for the unit sphere by Laarhoven (arXiv 2015) we use a kernel embedding technique by Rahimi & Recht (NIPS 2007) to show a solution to the $(r,cr)$-near neighbor problem in $\ell_s^d$-space for $0 < s \leq 2$ with query and update exponents $\rho_q=\frac{c^s(1+\lambda)^2}{(c^s+\lambda)^2}$ and $\rho_u=\frac{c^s(1-\lambda)^2}{(c^s+\lambda)^2}$ where $\lambda\in[-1,1]$ is a tradeoff parameter. This result improves upon the space-time tradeoff of Kapralov (PODS 2015) and is shown to be optimal in the case of a balanced tradeoff. Finally, we show a lower bound for the space-time tradeoff on the unit sphere that matches Laarhoven's and our own upper bound in the case of random data.