一种用于精度矩阵的替代图形Lasso算法
An Alternative Graphical Lasso Algorithm for Precision Matrices
AI总结：
本文提出一种基于精度矩阵最后一列新参数化的块坐标下降算法，将正则化对数似然分解为两个凸函数，实现与DP-GLasso相当的稀疏精度矩阵估计性能。
摘要：
图形Lasso（GLasso）算法在估计稀疏精度矩阵方面快速且被广泛使用（Friedman等人，2008）。在高维协方差估计文献中，其核心作用可与Lasso回归在均值向量稀疏估计中的作用相媲美。Mazumder和Hastie（2011）揭示、解决并呈现了关于其优化目标、收敛性、正定性和性能的一些谜团，从而引出了新的/改进的（对偶-原始）DP-GLasso算法。利用精度矩阵最后一列的新的且略有不同的重新参数化，我们证明正则化的正态对数似然自然分解为两个易于最小化的凸函数之和，其中一个为Lasso回归问题。这种分解是开发一种透明、简单的迭代块坐标下降算法来计算GLasso更新的关键，其性能与DP-GLasso相当。特别是，我们的算法从一开始就以精度矩阵作为其优化目标，并保留了DP-GLasso算法的所有有利性质。
英文摘要：
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.
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