A Constrained L1 Minimization Approach to Sparse Precision Matrix Estimation

TL;DR

This paper introduces a constrained L1 minimization method for estimating sparse inverse covariance matrices, demonstrating its theoretical properties, convergence rates, and practical effectiveness in graphical model selection and real data analysis.

Contribution

It proposes a new L1 minimization approach for sparse precision matrix estimation with proven convergence rates and practical implementation via linear programming.

Findings

Achieves convergence rate of s√(log p/n) under spectral norm.

Performs well in simulated and real data, including breast cancer analysis.

Outperforms existing methods in numerical experiments.

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 $s\sqrt{\log p/n}$ 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.