How well can we estimate a sparse vector?
  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. 
Emmanuel J. Cand\`es and Mark A. Davenport
cs.IT math.IT math.ST stat.TH