Algorithms and Lower Bounds for Sparse Recovery

We consider the following k-sparse recovery problem: design a distribution of m× n matrix A, such that for any signal x, given Ax with high probability we can efficiently recover x̂ satisfying ‖x− x̂‖1 ≤ C mink-sparse x′ ‖x− x‖1. It is known that there exist such distributions with m = O(k log(n/k)) rows; in this thesis, we show that this bound is tight. We also introduce the set query algorithm, a primitive useful for solving special cases of sparse recovery using less than Θ(k log(n/k)) rows. The set query algorithm estimates the values of a vector x ∈ 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 ≤ ‖x− xS‖2 with probability 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: • Improve the sparse recovery of Zipfian distributions O(k log n) measurements from a 1 + approximation to a 1 + o(1) approximation, giving the first such approximation when k ≤ O(n1− ). • Recover block-sparse vectors with O(k) space and a 1 + approximation. Previous algorithms required either ω(k) space or ω(1) approximation. Thesis Supervisor: Piotr Indyk Title: Associate Professor