Abstract

We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices or require this information be known a priori. The framework proposed in this article allows for simultaneous estimation of the precision matrices and relationships between the precision matrices. Sparse and nonsparse estimators are proposed, both of which require solving a nonconvex optimization problem. To compute our proposed estimators, we use an iterative algorithm which alternates between a convex optimization problem solved by blockwise coordinate descent and a k-means clustering problem. Blockwise updates for the sparse estimator require computing an elastic net penalized precision matrix estimation problem, which we solve using a proximal gradient descent algorithm. We prove that this subalgorithm has a linear rate of convergence. In simulation studies and two real data applications, we show that our method can outperform competitors that ignore relevant relationships between precision matrices and performs similarly to methods which use prior information often unknown in practice. Supplementary materials for this article are available online.

摘要

我们提出了一个惩罚似然框架，用于估计来自不同类别的多个精度矩阵。大多数现有方法要么不包含有关精度矩阵之间关系的信息，要么要求先验地知道这些信息。本文中提出的框架允许同时估计精度矩阵和精度矩阵之间的关系。提出了稀疏和非稀疏估计器，两者都需要解决非凸优化问题。为了计算我们提出的估计量，我们使用了一种迭代算法，该算法在通过分块坐标下降解决的凸优化问题和k-意味着聚类问题。稀疏估计器的逐块更新需要计算弹性网络惩罚精度矩阵估计问题，我们使用近端梯度下降算法解决该问题。我们证明该子算法具有线性收敛速度。在模拟研究和两个真实数据应用中，我们表明我们的方法可以胜过忽略精度矩阵之间相关关系的竞争对手，并且与使用实践中通常未知的先验信息的方法类似。本文的补充材料可在线获取。