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 ℓ1 estimation procedure such as the Dantzig selector. In this sense, compressive sensing techniques cannot essentially be improved. © 2012 Elsevier Inc.
Candès, E. J., & Davenport, M. A. (2013). How well can we estimate a sparse vector? Applied and Computational Harmonic Analysis. Academic Press Inc. https://doi.org/10.1016/j.acha.2012.08.010