Journal Article

A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation

Cai, Tony, Liu, Weidong, Luo, Xi

2011

Overview

This article proposes a constrained ℓ 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 when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise ℓ ∞ 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.

• Publisher: Taylor & Francis,American Statistical Association
• Subject: Acceleration of convergence / Analytical estimating / Applications / Consistent estimators / Covariance / Covariance matrices / Covariance matrix / Datasets / Estimation methods / Estimation theory / Estimators / Exact sciences and technology / Frobenius norm / Gaussian graphical model / General topics / Mathematics / Numerical analysis / Numerical analysis. Scientific computation / Numerical linear algebra / Precision matrix / Probability and statistics / Rate of convergence / Sciences and techniques of general use / Spectral norm / Statistical estimation / Statistics / Theory and Methods / Threshing