Graphical Lasso

The graphical Lasso estimates a sparse precision matrix by minimizing a Gaussian negative log-likelihood plus an entrywise ${\ell }^1$ penalty.

For objective $-\log \det \Omega +\mathrm{Tr}(S\Omega )+\lambda \Vert \Omega {\Vert }_{1,\mathrm{entry}}$, the KKT conditions are $-{\Omega }^{-1}+S+\lambda Z=0$ with $Z_{ij}\in \partial |{\Omega }_{ij}|$.