Reviews: Minimax Estimation of Bandable Precision Matrices

Summary The paper establishes the first theoretical guarantees for statistical estimation of bandable precision matrices. The authors propose an estimator for the precision matrix given a set of independent observations, and show that this estimator is minimax optimal. Interestingly the minimax rate for estimating banded precision matrices is shown to be equal to the corresponding rate for estimating banded covariance matrices. The upper bound follows by studying the behavior of inverses of small blocks along the diagonal of the covariance matrix together with classical random matrix theory results. The lower bound follows by constructing two specially defined subparameter spaces and applying testing arguments.