中国科学技术大学郭潇:L1处罚的精密矩阵条件数量的限制估计的影响

 

　　中国科学技术大学有数学、物理学、化学、地球物理学、生物学、科学技术史、力学、仪器科学与技术、材料科学与工程、动力工程及工程热物理、电子科学与技术、信息与通信工程、控制科学与工程、计算机科学与技术、核科学与技术、环境科学与工程、生物医学工程、管理科学与工程、工商管理、公共管理、软件工程、安全科学与工程、统计学在职研究生、生态学、地质学、天文学、哲学、应用经济学、新闻传播学、法学、大气科学、光学工程一级学科硕士学位授权点。

 

　　大型精密矩阵估计是高维推论的基础。一个重要的问题是要处理的精度矩阵的估计，通常是有限的样品中遇到的病态，但在文献中很少影响。在本文中，我们专注于通过征收界的估计，从而有效地保证了精心调理的条件数估计精度矩阵。具体来说，我们建议基于相关的估计，既条件数和L1惩罚约束，产生具有收敛理论上保证率精度矩阵估计。这一结果进一步使我们能够证明，结合了L1惩罚是必要的实现典型的高维设置所产生的估计的一致性，而当L1罚不存在将出现不一致的情况。基于乘数的交替方向法的算法开发来实现所提出的方法，揭示了模拟研究，表现令人满意。该方法以呼叫中心数据的应用说明。

 

　　原文：Estimation of large precision matrices is fundamental to high-dimensional inference. An important issue is to deal with ill-conditioning of the precision matrix estimate, typically encountered in finite-samples, but was rarely studied in the literature. In this paper, we focus on estimating the precision matrix by imposing a bound on the condition number of the estimate, which effectively ensures well-conditioning. Specifically, we propose a correlation-based estimator, constrained with both the condition number and the L1 penalty, yielding a precision matrix estimator with theoretically guaranteed rate of convergence. This result further enables us to demonstrate that incorporating the L1 penalty is necessary for achieving consistency of the resulting estimator in typical high-dimensional settings, while inconsistency will occur when the L1 penalty is absent. An algorithm based on the alternating direction method of multipliers is developed to implement the proposed method, which reveals the satisfactory performance in simulation studies. An application of the method to a call center data is illustrated.