Efficient Sketches for the Set Query Problem

TL;DR

This paper introduces an efficient randomized sparse binary linear sketching algorithm for estimating vector values over a support set, with applications in sparse recovery and heavy hitter detection.

Contribution

It presents a novel sketching method that achieves near-optimal space and approximation guarantees for sparse vector recovery problems.

Findings

Achieves O(k) recovery time with high probability.

Improves heavy hitter recovery to (1+o(1)) approximation in O(k log n) space.

Enables block-sparse vector recovery with near-optimal space and approximation.

Abstract

We develop an algorithm for estimating the values of a vector x in 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 ||x' - x_S||_2 <= eps ||x - x_S||_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+eps) approximation to a (1 + o(1)) approximation, giving the first such approximation in O(k log n) space when k <= O(n^{1-eps}). 2. Recover block-sparse vectors with O(k) space and a (1+eps) approximation. Previous algorithms required either omega(k) space or omega(1) approximation.