Scalable inference for high-dimensional precision matrix

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.