Lower Bounds for Sparse Recovery

TL;DR

This paper proves that the known optimal measurement bound for sparse recovery matrices is tight, even in randomized settings, establishing fundamental limits for compressed sensing.

Contribution

It establishes a tight lower bound of O(k log(n/k)) on the number of measurements needed for sparse recovery, matching existing upper bounds.

Findings

The bound holds even for randomized measurement matrices.

The result confirms the optimality of current sparse recovery methods.

The proof applies to algorithms with constant probability success.

Abstract

We consider the following k-sparse recovery problem: design an m x n matrix A, such that for any signal x, given Ax we can efficiently recover x' satisfying ||x-x'||_1 <= C min_{k-sparse} x"} ||x-x"||_1. It is known that there exist matrices A with this property that have only O(k log (n/k)) rows. In this paper we show that this bound is tight. Our bound holds even for the more general /randomized/ version of the problem, where A is a random variable and the recovery algorithm is required to work for any fixed x with constant probability (over A).