  A constrained L1 minimization method is proposed for estimating a sparse
inverse covariance matrix based on a sample of $n$ iid $p$-variate random
variables. The resulting estimator is shown to enjoy a number of desirable
properties. In particular, it is shown that the rate of convergence between the
estimator and the true $s$-sparse precision matrix under the spectral norm is
$s\sqrt{\log p/n}$ when the population distribution has either exponential-type
tails or polynomial-type tails. Convergence rates under the elementwise
$L_{\infty}$ norm and Frobenius norm are also presented. In addition, graphical
model selection is considered. The procedure is easily implementable by linear
programming. Numerical performance of the estimator is investigated using both
simulated and real data. In particular, the procedure is applied to analyze a
breast cancer dataset. The procedure performs favorably in comparison to
existing methods.
