基于L_1范数最小化的逆协方差矩阵估计

A approach to Precision Matrix Estimation Based on L_1 Norm Minimization

由于在高维空间中,基于固定维数的经典方法和结果不再适用,样本协方差矩阵不可逆,估计逆协方差矩阵时存在不稳定、计算成本高和非精确等问题,提出了一种L1范数最小化方法来有效估计高维逆协方差矩阵即精确矩阵.当总体分布满足指数类型条件或者多项式类型条件时,所提估计方法在各种范数下的收敛速率优于其他现存的方法.经分析验证,所提方法为凸优化问题,可采用交替方向乘子算法来解决.之后通过R语言在模拟数据和实际数据下进行仿真分析,并与Glasso方法对比逆协方差的估计性能和图恢复性能,结果表明所提估计方法准确率高、计算成本低.最后,将所提估计方法用来分析白血病数据集,并运用聚类分析对白血病人进行分类.

Because it is irreversible for sample covariance matrix in high-dimensional situation,and it is not stable to estimate the inverse covariance matrix.What＇s more,the classic methods and results based on fixed dimension is no longer applicable.AL1 norm minimization method was proposed to estimate a high-dimensional inverse covariance matrix,and the related problem namely Gaussian graphical model selection were analyzed.The convergence rates under the various norm were given when the population distribution has either exponential-type tails or polynomial-type tails.The convergence rates are superior to other existing methods.The methods is convex optimization problem,and can be converted into linear programming,then apply alternating direction method of multiplier algorithm to solve it.Numerical performance of the estimator was investigated using both simulated and real data by R language.The precision matrix estimation performance and recovery performance of the various models was compared.The results show that the proposed method has high accuracy,low computational cost and rapid running speed.In addition,the procedure was applied to analyze a Leukemia dataset and used cluster analysis to classify the patients.