= Graphical Lasso
{c}
{title2=$\widehat\Omega_\lambda$}
{wiki}
The <Graphical Lasso> estimates a <sparse matrix> for the <precision matrix> by minimizing a penalized Gaussian objective over <positive-definite matrices>. With a <sample covariance matrix> $S$ and $\lambda>0$, the all-entry penalty convention is
$$
\widehat\Omega=\operatorname*{argmin}_{\Omega\succ0}\left\{-\log\det\Omega+\operatorname{tr}(S\Omega)+\lambda\sum_{i,j}|\Omega_{ij}|\right\}.
$$
The <entrywise matrix L1 norm> encourages zero entries, while the domain enforces <positive definiteness>. In a <Gaussian graphical model>, zero off-diagonal <precision matrix> entries encode <conditional independence> given the other coordinates.<Sparse inverse covariance estimation with the graphical lasso by Jerome H. Friedman and others>[Equation 2.1 and graphical interpretation]{ref}
The penalty convention matters. Writing $W=\widehat\Omega^{-1}$, the <Graphical-Lasso Karush-Kuhn-Tucker conditions> give $W_{ii}=S_{ii}+\lambda$ when diagonal entries are penalized, but $W_{ii}=S_{ii}$ when they are excluded.<Sparse inverse covariance estimation with the graphical lasso by Jerome H. Friedman and others>[Equation 2.6 and Remark 2.2]{ref} For example, diagonal $S=\operatorname{diag}(1,4)$ and $\lambda=1$ give $\widehat\Omega=\operatorname{diag}(1/2,1/5)$ under the all-entry convention.
The <Graphical Lasso column regression> relates its <matrix> optimization to <Lasso> subproblems. This is different from independently penalizing a raw <covariance> inverse, and the <CLIME precision-matrix estimator> uses a different constrained objective.
Back to article page
