Fast and adaptive sparse precision matrix estimation in high dimensions

• We propose a new procedure for sparse precision matrix estimation.
• We are among the first to establish the theory of cross validation for this problem.
• The conditions are slightly weaker than an important penalized likelihood method.
• Improved numerical performance is observed in several examples.

This paper proposes a new method for estimating sparse precision matrices in the high dimensional setting. It has been popular to study fast computation and adaptive procedures for this problem. We propose a novel approach, called Sparse Column-wise Inverse Operator, to address these two issues. We analyze an adaptive procedure based on cross validation, and establish its convergence rate under the Frobenius norm. The convergence rates under other matrix norms are also established. This method also enjoys the advantage of fast computation for large-scale problems, via a coordinate descent algorithm. Numerical merits are illustrated using both simulated and real datasets. In particular, it performs favorably on an HIV brain tissue dataset and an ADHD resting-state fMRI dataset.