  Compressed sensing deals with the reconstruction of sparse signals using a
small number of linear measurements. One of the main challenges in compressed
sensing is to find the support of a sparse signal. In the literature, several
bounds on the scaling law of the number of measurements for successful support
recovery have been derived where the main focus is on random Gaussian
measurement matrices. In this paper, we investigate the noisy support recovery
problem from an estimation theoretic point of view, where no specific
assumption is made on the underlying measurement matrix. The linear
measurements are perturbed by additive white Gaussian noise. We define the
output of a support estimator to be a set of position values in increasing
order. We set the error between the true and estimated supports as the
$\ell_2$-norm of their difference. On the one hand, this choice allows us to
use the machinery behind the $\ell_2$-norm error metric and on the other hand,
converts the support recovery into a more intuitive and geometrical problem.
First, by using the Hammersley-Chapman-Robbins (HCR) bound, we derive a
fundamental lower bound on the performance of any \emph{unbiased} estimator of
the support set. This lower bound provides us with necessary conditions on the
number of measurements for reliable $\ell_2$-norm support recovery, which we
specifically evaluate for uniform Gaussian measurement matrices. Then, we
analyze the maximum likelihood estimator and derive conditions under which the
HCR bound is achievable. This leads us to the number of measurements for the
optimum decoder which is sufficient for reliable $\ell_2$-norm support
recovery. Using this framework, we specifically evaluate sufficient conditions
for uniform Gaussian measurement matrices.
