Sparse Covariance Estimation from Quadratic Measurements: A Precise Analysis

We study the problem of estimating a high-dimensional sparse covariance matrix, Σ0, from a finite number of quadratic measurements, i.e., measurements ${\text{a}}_i^T{\Sigma _0}{{\text{a}}_i}$ which are quadratic forms in the measurement vectors ai resulting from the covariance matrix, Σ0. Such a problem arises in applications where we can only make energy measurements of the underlying random variables. We study a simple LASSO-like convex recovery algorithm which involves a squared 2-norm (to match the covariance estimate to the measurements), plus a regularization term (that penalizes the ℓ1−norm of the non-diagonal entries of Σ0 to enforce sparsity). When the measurement vectors are i.i.d. Gaussian, we obtain the precise error performance of the algorithm (accurately determining the estimation error in any metric, e.g., 2-norm, operator norm, etc.) as a function of the number of measurements and the underlying distribution of Σ0. In particular, in the noiseless case we determine the necessary and sufficient number of measurements required to perfectly recover Σ0 as a function of its sparsity. Our results rely on a novel comparison lemma which relates a convex optimization problem with "quadratic Gaussian" measurements to one which has i.i.d. Gaussian measurements.