A Constrained l₁ Minimization Approach to Sparse Precision Matrix Estimation

Tony Cai, Weidong Liu and Xi Luo

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• Abstract: 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(log p/n)^1/2 when the population distribution has either exponential-type tails or polynomial-type tails. Convergence rates under the elementwise l₁ 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.

• Paper: pdf file.

• R Package for the CLIME estimator. Here is the Clime-Manual file.

• Other related papers:

  Cai, T. & Zhou, H. (2009).
  Minimax estimation of large covariance matrices under l₁ norm (with discussion)
  Statistica Sinica, to appear.

  Cai, T., Zhang, C.-H. & Zhou, H. (2010).
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  Cai, T., Wang, L. & Xu, G. (2010).
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