The Sparsity-Constrained Graphical Lasso

This paper introduces the Sparsity-constrained Graphical Lasso (SCGlasso) for the precision matrix, \(\mathbf {\Theta }\) , in a multivariate Gaussian framework. The estimator is designed to produce a shrunk estimate of \(\mathbf {\Theta }\) , while simultaneously imposing a certain degree of sparsity, which is crucial for reconstructing the conditional dependence graph and the partial correlation graph. The proposed method employs an \(\ell _1\) -norm (Glasso) regularization to achieve shrinkage and imposes an \(\ell _0\) -pseudo-norm constraint to ensure sparsity. The proposed approach performs well compared to Glasso on simulated data, also in contexts where the number of variables p exceeds the number of observations n.