Efficient minimax estimation of a class of high-dimensional sparse precision matrices

Estimation of the covariance matrix and its inverse, the precision matrix, in high-dimensional situations is of great interest in many applications. In this paper, we focus on the estimation of a class of sparse precision matrices which are assumed to be approximately inversely closed for the case that the dimensionality p can be much larger than the sample size n, which is fundamentally different from the classical case that p < n. Different in nature from state-of-the-art methods that are based on penalized likelihood maximization or constrained error minimization, based on the truncated Neumann series representation, we propose a computationally efficient precision matrix estimator that has a computational complexity of O(p ³). We prove that the proposed estimator is consistent in probability and in L ² under the spectral norm. Moreover, its convergence is shown to be rate-optimal in the sense of minimax risk. We further prove that the proposed estimator is model selection consistent by establishing a convergence result under the entry-wise ∞-norm. Simulations demonstrate the encouraging finite sample size performance and computational advantage of the proposed estimator. The proposed estimator is also applied to a real breast cancer data and shown to outperform existing precision matrix estimators.