Efficient sketches for the set query problem

Abstract

We develop an algorithm for estimating the values of a vector a; € R" over a support S of size k from a randomized sparse binary linear sketch Ax of size O(k). Given Ax and S, we can recover x′ with ∥x′-xs∥2≤ε∥x;-xs∥2 with probabiUty at least 1 -k-Ω(1). The recovery takes O(k) time. While interesting in its own right, this primitive also has a number of applications. For example, we can: 1. Improve the linear k-sparse recovery of heavy hitters in Zipfian distributions with O(k logn) space from a 1 + ε approximation to a 1 + o(l) approximation, giving the first such approximation in O(k logn) space when l≤ O(n1-ε) 2. Recover block-sparse vectors with O(k) space and a l+ε approximation. Previous algorithms required either w(k) space or w(l) approximation.