The Graphical Lasso (GLasso) algorithm is fast and widely used for estimating sparse precision matrices (Friedman et al., 2008). Its central role in the literature of high-dimensional covariance estimation rivals that of Lasso regression for sparse estimation of the mean vector. Some mysteries regarding its optimization target, convergence, positive-definiteness and performance have been unearthed, resolved and presented in Mazumder and Hastie (2011), leading to a new/improved (dual-primal) DP-GLasso. Using a new and slightly different reparametriztion of the last column of a precision matrix we show that the regularized normal log-likelihood naturally decouples into a sum of two easy to minimize convex functions one of which is a Lasso regression problem. This decomposition is the key in developing a transparent, simple iterative block coordinate descent algorithm for computing the GLasso updates with performance comparable to DP-GLasso. In particular, our algorithm has the precision matrix as its optimization target right at the outset, and retains all the favorable properties of the DP-GLasso algorithm.

翻译：图套索（Graphical Lasso, GLasso）算法是一种快速且广泛用于估计稀疏精度矩阵的方法（Friedman 等，2008）。它在高维协方差估计文献中的核心地位可与套索回归在均值向量稀疏估计中的作用相媲美。关于其优化目标、收敛性、正定性及性能的一些谜团已在 Mazumder 和 Hastie（2011）的研究中被揭示、解决并呈现，进而提出了一种新的/改进的（对偶-原始）DP-GLasso 算法。通过采用一种新颖且略有不同的精度矩阵最后一列重参数化方法，我们证明正则化正态对数似然自然解耦为两个易于最小化的凸函数之和，其中一个即为套索回归问题。这一分解是开发一种透明且简单的迭代块坐标下降算法来计算 GLasso 更新的关键，其性能可与 DP-GLasso 相媲美。特别地，我们的算法从一开始就明确将精度矩阵作为优化目标，并保留了 DP-GLasso 算法的所有优良性质。