  We introduce a new sparse estimator of the covariance matrix for
high-dimensional models in which the variables have a known ordering. Our
estimator, which is the solution to a convex optimization problem, is
equivalently expressed as an estimator which tapers the sample covariance
matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of
this adaptivity, the convex banding estimator enjoys theoretical optimality
properties not attained by previous banding or tapered estimators. In
particular, our convex banding estimator is minimax rate adaptive in Frobenius
and operator norms, up to log factors, over commonly-studied classes of
covariance matrices, and over more general classes. Furthermore, it correctly
recovers the bandwidth when the true covariance is exactly banded. Our convex
formulation admits a simple and efficient algorithm. Empirical studies
demonstrate its practical effectiveness and illustrate that our exactly-banded
estimator works well even when the true covariance matrix is only close to a
banded matrix, confirming our theoretical results. Our method compares
favorably with all existing methods, in terms of accuracy and speed. We
illustrate the practical merits of the convex banding estimator by showing that
it can be used to improve the performance of discriminant analysis for
classifying sound recordings.
