Precision matrix ($\Omega ={\Sigma }^{-1}$)

The precision matrix is the inverse of a nonsingular covariance matrix. For a multivariate normal distribution, a zero off-diagonal precision entry encodes conditional independence of the corresponding coordinates given all remaining coordinates.

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 $Z_{ij}\in \partial |{\Omega }_{ij}|$.