On Compressive Sensing of Sparse Covariance Matrices Using Deterministic Sensing Matrices

Abstract

This paper considers the problem of determining the sparse covariance matrix $\mathbf{X}$ of an unknown data vector $\pmb{x}$ by observing the covariance matrix $\mathbf{Y}$ of a compressive measurement vector $\pmb{y}=\mathbf{A}\pmb{x}$. We construct deterministic sensing matrices $\mathbf{A}$ for which the recovery of a $k$ -sparse covariance matrix $\mathbf{X}$ from $m$ values of $\mathbf{Y}$ is guaranteed with high probability. In particular, we show that the number of measurements $m$ scales linearly with the sparsity $k$.