Corrections to “Asymptotic Achievability of the Cramér–Rao Bound for Noisy Compressive Sampling” [Mar 09 1233-1236]

Given
$N$
noisy measurements denoted by
${\mathbf y}$
and an overcomplete Gaussian dictionary,
${\mathbf A}$
, the authors of
[1]
establish the existence and the asymptotic statistical efficiency of an unbiased estimator unaware of the locations of the nonzero entries, collected in set
$\mathcal {I}$
, in the deterministic
$L$
-sparse signal
${\mathbf x}$
. More precisely, there exists an estimator
${\hat{\mathbf {x}}}({\mathbf y}, {\mathbf A})$
unaware of set
$\mathcal {I}$
with a variance reaching the oracle-Cramér–Rao Bound in the asymptotic scenario, i.e., for
$N,L\rightarrow \infty$
and
$L/N \rightarrow \alpha \in (0,1)$
. As was noted in the paper “Fundamental limits and constructive methods for estimation and sensing of sparse signals” by B. Babadi, the existence proof remains true even though Lemma 3.5 and (20) the paper “Asymptotic achievability of the Cramer–Rao bound for noisy compressive sampling” are inexact. In this note, the exact closed-form expression of the variance of the estimator
${\hat{\mathbf {x}}}({\mathbf y}, {\mathbf A})$
is provided, and its practical usefulness is numerically illustrated with the orthogonal matching pursuit estimator.