稀疏精度矩阵估计的约束L1最小化方法
A Constrained L1 Minimization Approach to Sparse Precision Matrix Estimation
摘要：
提出了一种基于$n$ iid $p$ -变量随机变量样本估计稀疏逆协方差阵的约束L1最小化方法。由此产生的估计，享有一些理想的属性。特别是，它示出的估计和真正的$s$稀疏精度矩阵的谱范数下的收敛速度是$s\sqrt{\log p/n}$时，总体分布有指数型尾部或多项式型尾部。给出了在逐元素$L_{\infty}$范数和Frobenius范数下的收敛速度.此外，考虑了图形化的模型选择。该方法易于用线性规划实现。使用模拟和真实数据的估计的数值性能进行了研究。特别地，该过程被应用于分析乳腺癌数据集。与现有方法相比，该方法表现良好。
英文摘要：
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.
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