标题： 基于高维协方差估计的高斯图模型 显示英文标题

标题： High-dimensional covariance estimation based on Gaussian graphical models

摘要： 无向图常用于描述高维分布。 在稀疏性条件下，可以使用$\ell_1$惩罚方法估计图。 我们提出并研究以下方法。 我们将多重回归方法与阈值化和重拟的思想相结合：首先通过阈值化许多$\ell_1$范数惩罚回归函数中的每一个来推断稀疏的无向图模型结构；然后使用最大似然估计量估计协方差矩阵及其逆矩阵。 我们证明，在适当条件下，这种方法在图结构方面产生一致估计，并且在协方差矩阵及其逆矩阵的算子范数和Frobenius范数方面具有快速收敛速率。 我们还推导了Kullback Leibler散度的显式界。 显示英文摘要

摘要： Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using $\ell_1$-penalization methods. We propose and study the following method. We combine a multiple regression approach with ideas of thresholding and refitting: first we infer a sparse undirected graphical model structure via thresholding of each among many $\ell_1$-norm penalized regression functions; we then estimate the covariance matrix and its inverse using the maximum likelihood estimator. We show that under suitable conditions, this approach yields consistent estimation in terms of graphical structure and fast convergence rates with respect to the operator and Frobenius norm for the covariance matrix and its inverse. We also derive an explicit bound for the Kullback Leibler divergence.