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arXiv · 2607.07735

The Regularization Parameter: Sparse Precision Matrix Estimation

Aryan Eftekhari·Daniel Sergio Vega·Ernst-Jan Camiel Wit·Olaf Schenk
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Abstract

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.

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Aryan Eftekhari, Daniel Sergio Vega, Ernst-Jan Camiel Wit, Olaf Schenk. 2026-07-07. The Regularization Parameter: Sparse Precision Matrix Estimation. https://arxiv.org/abs/2607.07735

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