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In this paper, we propose a general framework for designing sensing matrix$\boldsymbol{A} \in \mathbb{R}^{d\times p}$, for estimation of sparsecovariance 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 $\boldsymbol{x} \in \mathbb{R}^p$. Byviewing covariance recovery as inference over factor graphs via message passingalgorithm, 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., equalimportance is given to all entries of the covariance, and (2) preferentialsensing, i.e., higher importance is given to a part of the covariance matrix.Through experiments, we show that the sensing matrix designed via densityevolution can match the state-of-the-art for covariance recovery in the regularsensing paradigm and attain improved performance in the preferential sensingregime. Additionally, we study the feasibility of causal graph structurerecovery using the estimated covariance matrix obtained from the compressedmeasurements.

A Density Evolution framework for Preferential Recovery of Covariance and Causal Graphs from Compressed Measurements