  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 $k
\times k$ 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.
