High-dimensional Sparse Precision Matrix Estimation via Sparse Column Inverse Operator

This paper proposes a new method for estimating sparse precision matrices in the high dimensional setting. This procedure applies a novel Sparse Column-wise Inverse Operator (SCIO) to modified sample covariance matrices. We establish the convergence rates of this procedure under various matrix norms. Under the Frobenius norm loss, we prove theoretical guarantees on using cross validation to pick data-driven tunning parameters. Another important advantage of this estimator is its efficient computation for large-scale problems, using a path-following coordinate descent algorithm we provide. Numerical merits of our estimator are also illustrated using simulated and real datasets. In particular, this method is found to perform favorably on analyzing an HIV brain tissue dataset and an ADHD resting fMRI dataset.