Efficient Sketches for the Set Query Problem

We develop an algorithm for estimating the values of a vector x is an element of R-n 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 parallel to x' - x(S)parallel to(2) <= epsilon parallel to x - x(S)parallel to(2) with probability at least 1 - k(-Omega(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 log n) space from a 1 + epsilon approximation to a 1 + o(1) approximation, giving the first such approximation in O(k log n) space when k <= O(n(1-epsilon)). 2. Recover block-sparse vectors with O(k) space and a 1 + epsilon approximation. Previous algorithms required either omega(k) space or omega(1) approximation.