标题： 鲁棒高维精度矩阵估计 显示英文标题

标题： Robust high-dimensional precision matrix estimation

摘要： 多变量数据的依赖结构可以通过协方差矩阵$\Sigma$进行分析。在许多领域中，精度矩阵$\Sigma^{-1}$更具信息量。由于样本协方差估计器在高维情况下是奇异的，因此无法用于获得精度矩阵估计器。一种流行的高维估计器是图形套索，但它缺乏鲁棒性。我们考虑高维独立污染模型。在这里，即使数据矩阵中有少量被污染的单元格，也可能导致大量被污染的行。传统鲁棒程序通过降低整个观测值的权重，这将导致信息损失。在本文中，我们正式证明用一个元素级鲁棒协方差矩阵替换图形套索中的样本协方差矩阵，可以得到一个元素级鲁棒、稀疏且在高维下可计算的精度矩阵估计器。给出了一些此类元素级鲁棒协方差估计器的例子。最终的精度矩阵估计器是正定的，在元素级污染下具有较高的崩溃点，并且可以快速计算。 显示英文摘要

摘要： The dependency structure of multivariate data can be analyzed using the covariance matrix $\Sigma$. In many fields the precision matrix $\Sigma^{-1}$ is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular high-dimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations, which is done by traditional robust procedures, would then results in a loss of information. In this paper, we formally prove that replacing the sample covariance matrix in the graphical lasso with an elementwise robust covariance matrix leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Examples of such elementwise robust covariance estimators are given. The final precision matrix estimator is positive definite, has a high breakdown point under elementwise contamination and can be computed fast.