# The Regularization Parameter: Sparse Precision Matrix Estimation

arXiv: https://arxiv.org/abs/2607.07735 · HTML: https://mlsanity.com/paper/2607.07735/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 50 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [On the folded normal distribution](https://mlsanity.com/paper/1402.3559/) (2014)
- [Approximate Cross-validation: Guarantees for Model Assessment and Selection](https://mlsanity.com/paper/2003.00617/) (2020)
- [Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation](https://mlsanity.com/paper/1306.3212/) (2013)
- [The huge Package for High-dimensional Undirected Graph Estimation in R](https://mlsanity.com/paper/2006.14781/) (2020)
- [The flare Package for High Dimensional Linear Regression and Precision Matrix Estimation in R](https://mlsanity.com/paper/2006.15419/) (2020)
- [A computationally fast alternative to cross-validation in penalized Gaussian graphical models](https://mlsanity.com/paper/1309.6216/) (2013)
- [Brain modularity controls the critical behavior of spontaneous activity](https://mlsanity.com/paper/1403.6339/) (2014)

## Abstract (arXiv)

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 ℓ1-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.