A constrained \(\ell _{1}\) minimization approach to sparse precision matrix estimation. (English) Zbl 1232.62087

Cai, Tony; Liu, Weidong; Luo, Xi

J. Am. Stat. Assoc. 106, No. 494, 594-607 (2011).

Summary: This article proposes a constrained \(\ell _{1}\) 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 \(s \sqrt {\log p/n}\) when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise \(\ell _{\infty }\) 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.

MSC:

62H12	Estimation in multivariate analysis
62F12	Asymptotic properties of parametric estimators
90C05	Linear programming
62P10	Applications of statistics to biology and medical sciences; meta analysis
65C60	Computational problems in statistics (MSC2010)

Keywords:

covariance matrix; Frobenius norm; Gaussian graphical model; rate of convergence; spectral norm