We consider the problem of recovering the common support of a set of
$k$-sparse signals $\{\mathbf{x}_{i}\}_{i=1}^{L}$ from noisy linear
underdetermined measurements of the form
$\{{\Phi} \mathbf{x}_{i}+\mathbf{w}_{i}\}_{i=1}^{L}$ where
${\Phi}\in\rr^{m\times N}$ $(m<N)$ is the sensing matrix and
$\mathbf{w}_{i}$ is the additive noise. We employ a Bayesian setup where we impose a Gaussian prior with zero mean and a
common diagonal covariance matrix $\mathbf{\Gamma}$ across all
$\mathbf{x}_{i},$ and formulate the support recovery problem as one of
covariance estimation.
We develop an algorithm to find the
approximate maximum-likelihood estimate of $\mathbf{\Gamma}$ using a
modified reweighted minimization procedure. Empirically, we
find that the proposed algorithm succeeds in exactly recovering
the common support with high probability in the $k<m$ regime with $L$
of the order of $m$ and in the $k\ge m$ regime with larger $L$. The key advantage of the proposed algorithm is that its complexity is independent of $L$, unlike existing sparse support recovery algorithms.