Minimax Estimation of Bandable Precision Matrices
The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, which encodes conditional independence statements between variables. Minimax results under the spectral norm have previously been established for covariance matrices, both sparse and banded, and for sparse precision matrices. We establish minimax estimation bounds for estimating banded precision matrices under the spectral norm. Our results greatly improve upon the existing bounds; in particular, we find that the minimax rate for estimating banded precision matrices matches that of estimating banded covariance matrices. The key insight in our analysis is that we are able to obtain barely-noisy estimates of k×k subblocks of the precision matrix by inverting slightly wider blocks of the empirical covariance matrix along the diagonal. Our theoretical results are complemented by experiments demonstrating the sharpness of our bounds.
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Summary
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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. The authors support their theoretical development via representative numerical results.
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Strengths
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Clarity: The paper is very nicely written and the authors appropriately convey the ideas and relevance of their work.
Novelty: While I am not too familiar with the surrounding literature, the ideas in this paper appear novel. In particular, the estimator is both intuitive, as well as (relatively) easy to compute via the tricks that the authors discuss in Section 2.2. The techniques used to prove both upper and lower bounds are interesting and sound. 
Significance: The authors establish matching upper and lower bounds, thereby effectively resolving the question of bandable precision matrix estimation (at least in the context of Gaussian random variables).
Typo: Should it be F_alpha in (3), not P_alpha?
Update after authors' response: Good paper; no further comments to add.