The Regularization Parameter: Sparse Precision Matrix Estimation

• URL: http://arxiv.org/abs/2607.07735v1
• Date: Tue, 07 Jul 2026 13:19:19 GMT
• Title: The Regularization Parameter: Sparse Precision Matrix Estimation
• Abstract summary: Sparse precision matrix estimation provides an efficient framework for modeling conditional dependencies in high-dimensional, low-sample-size data.<n>A recurring challenge is appropriately selecting the regularization parameter that controls estimator sparsity.<n>We propose a closed-form, matrix-valued regularization parameter derived from the sampling distribution of the first-order optimality conditions.
• Score: 1.3532832187980637
• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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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