A Framework For Similarity Search With Space-time Tradeoffs Using Locality-sensitive Filtering
Christiani Tobias. Arxiv 2016
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.
Similar Work