We first develop a lower bound technique that is particularly well suited for treating "two-directional" problems such as estimating sparse covariance matrices. The result can be viewed as a generalization of Le Cam's method in one direction and Assouad's Lemma in another. This lower bound technique is of independent interest and can be used for other matrix estimation problems.

We then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique. A thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm. The results are then extended to the general matrix l_w operator norms for 1 ≤ w ≤ ∞. In addition, we give a unified result on the minimax rate of convergence for sparse covariance matrix estimation under a class of Bregman divergence losses.

Cai, T., Zhang, C.-H. & Zhou, H. (2010).
Optimal rates of convergence for covariance matrix estimation
The Annals of Statistics 38, 2118-2144.

Cai, T., Liu, W. & Luo, X. (2011).
A constrained l₁ minimization approach to sparse precision matrix estimation.
J. American Statistical Association 106, 594-607.

Cai, T. & Liu, W. (2011).
Adaptive thresholding for sparse covariance matrix estimation.
J. American Statistical Association 106, 672-684.

Cai, T. & Zhou, H. (2012).
Minimax estimation of large covariance matrices under l₁ norm (with discussion).
Statistica Sinica 22, 1319-1378.

Cai, T. & Yuan, M. (2012).
Adaptive covariance matrix estimation through block thresholding.
The Annals of Statistics 40, 2014-2042.

Cai, T., Ren, Z. & Zhou, H. (2013).
Optimal rates of convergence for estimating Toeplitz covariance matrices.
Probability Theory and Related Fields 156, 101-143.