An $$\ell _0$$-constrained and $$\ell _1$$-regularized estimator for graphical models

Graphical models provide a powerful framework for representing conditional dependencies among random variables, where the precision matrix  encodes these relationships. Traditional approaches like the Graphical Lasso (Glasso), which employs l1 -regularization, promote sparsity but suffer from drawbacks such as biased estimates, sensitivity to regularization parameters, and weak control over false positive edges. Conversely, l0 -based methods enforce exact sparsity, but they are either continuous surrogates of the best-subset selector, that performs poorly outside low-dimensional settings; or iterative algorithms enforcing the l0-constraint directly, which do not immediately guarantee the preservation of positive definiteness in the domain of graphical models. To address these limitations, we introduce the Sparsity-Constrained Graphical Lasso (SCGlasso), a novel estimator that integrates an l0-constraint with an l1-penalty to balance sparsity and shrinkage effectively. We propose an efficient coordinate descent algorithm and establish its computational complexity and convergence to a local minimum. Through extensive simulations, we benchmark SCGlasso against state-of-the-art methods, including Glasso, Gslope, Selo, Atan and Exponential penalties. Our results demonstrate that SCGlasso achieves a competitive performance in both estimation and model selection accuracy, especially in low-sample regimes. Finally, we illustrate its practical utility on two empirical applications: gene expression data from Arabidopsis thaliana and financial returns of the Euro Stoxx 50 constituents, where SCGlasso uncovers interpretable network structures that competing methods often fail to recover.