  In this paper, we propose a general framework for designing sensing matrix
$\boldsymbol{A} \in \mathbb{R}^{d\times p}$, for estimation of sparse
covariance matrix from compressed measurements of the form $\boldsymbol{y} =
\boldsymbol{A}\boldsymbol{x} + \boldsymbol{n}$, where $\boldsymbol{y},
\boldsymbol{n} \in \mathbb{R}^d$, and $\boldsymbol{x} \in \mathbb{R}^p$. By
viewing covariance recovery as inference over factor graphs via message passing
algorithm, ideas from coding theory, such as \textit{Density Evolution} (DE),
are leveraged to construct a framework for the design of the sensing matrix.
The proposed framework can handle both (1) regular sensing, i.e., equal
importance is given to all entries of the covariance, and (2) preferential
sensing, i.e., higher importance is given to a part of the covariance matrix.
Through experiments, we show that the sensing matrix designed via density
evolution can match the state-of-the-art for covariance recovery in the regular
sensing paradigm and attain improved performance in the preferential sensing
regime. Additionally, we study the feasibility of causal graph structure
recovery using the estimated covariance matrix obtained from the compressed
measurements.
