A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation

T. Tony Cai, Weidong Liu, Xi Luo

Abstract:

This article proposes a constrained 1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid
p-variate random variables. The resulting estimator is shown to have a number of desirable properties. In particular, the rate of convergence
between the estimator and the true s-sparse precision matrix under the spectral norm is s
√log p/n when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise ∞ norm and Frobenius norm.
In addition, we consider graphical model selection. The procedure is easily implemented 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
and is found to perform favorably compared with existing methods.