The Regularization Parameter: Sparse Precision Matrix Estimation

Sparse precision matrix estimation provides an interpretable and computationally efficient framework for modeling conditional dependencies in high-dimensional, low-sample-size data.

A recurring challenge is appropriately selecting the regularization parameter that controls estimator sparsity and strikes a balance between underfitting and overfitting.

We propose a closed-form, matrix-valued regularization parameter derived from the sampling distribution of the first-order optimality conditions of the $\ell_1$-regularized Gaussian maximum-likelihood estimator.

By prescribing the probability that each nonzero entry of the estimator satisfies its optimality condition under resampling, we eliminate the need for cross-validation.

The resulting regularization parameter is shown to attain asymptotic scaling properties that, under standard conditions, provide consistency and sparsistency of the estimator.

On synthetic Gaussian and non-Gaussian datasets, as well as real-world gene microarray and neuroimaging applications, the proposed approach achieves estimation accuracy comparable to cross-validation, delivers superior support recovery, and reduces runtime by several orders of magnitude.