How well can we estimate a sparse vector?

TL;DR

This paper establishes a fundamental lower bound on the mean-squared error for estimating sparse vectors, showing that existing compressive sensing methods are nearly optimal and cannot be significantly improved.

Contribution

It provides a universal lower bound on estimation error that applies regardless of sensing matrix or estimation method, confirming the near-optimality of current compressive sensing techniques.

Findings

Lower bound on mean-squared error matches upper bounds from existing methods

Compressive sensing techniques are nearly optimal for sparse vector estimation

The bound holds regardless of sensing matrix or estimation procedure

Abstract

The estimation of a sparse vector in the linear model is a fundamental problem in signal processing, statistics, and compressive sensing. This paper establishes a lower bound on the mean-squared error, which holds regardless of the sensing/design matrix being used and regardless of the estimation procedure. This lower bound very nearly matches the known upper bound one gets by taking a random projection of the sparse vector followed by an $\ell_1$ estimation procedure such as the Dantzig selector. In this sense, compressive sensing techniques cannot essentially be improved.