Abstract

Statistical inference for precision matrix is of fundamental importance nowadays for learning conditional dependence structure in high-dimensional graphical models. Despite the fast growing literature, how to develop scalable inference with insensitive tuning of the regularization parameters still remains unclear in high dimensions. In this paper, we develop a new method called the graphical constrained projection inference (GCPI) to test individual entry of the precision matrix in a scalable and efficient way. The proposed test statistics are based on the constrained projection space yielded by certain screening procedures, which combine the strengths of the constrained projection and the screening procedures, thus enjoying the scalability and the tuning free property inherited from the above two methods. Theoretically, we prove that the new statistics enjoy the asymptotic normality and achieve the exact inference. Both numerical results and real data analysis confirm the advantage of our method.

摘要

精确矩阵的统计推断对于学习高维图模型中的条件依赖结构具有重要意义。尽管文献快速增长，但如何通过对正则化参数的不敏感调整来开发可扩展的推理在高维度上仍然不清楚。在本文中，我们开发了一种称为图形约束投影推理 (GCPI) 的新方法，以可扩展且有效的方式测试精度矩阵的各个条目。所提出的测试统计是基于某些筛选程序产生的约束投影空间，结合了约束投影和筛选程序的优点，从而享受了从上述两种方法继承的可扩展性和无调整特性。理论上，我们证明了新的统计量具有渐近正态性并实现了精确的推断。数值结果和实际数据分析都证实了我们方法的优势。