{"ID":6268106,"CreatedAt":"2026-07-10T01:11:38.759438437Z","UpdatedAt":"2026-07-10T13:03:38.548899896Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07735","arxiv_id":"2607.07735","title":"The Regularization Parameter: Sparse Precision Matrix Estimation","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.","short_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 betwe...","url_abs":"https://arxiv.org/abs/2607.07735","url_pdf":"https://arxiv.org/pdf/2607.07735v1","authors":"[\"Aryan Eftekhari\",\"Daniel Sergio Vega\",\"Ernst-Jan Camiel Wit\",\"Olaf Schenk\"]","published":"2026-07-07T13:19:19Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}