Paired group Lasso for Gaussian graph estimation (${\sum }_{j<k}\sqrt{{\Theta }_{jk}^2+{\Theta }_{kj}^2}$)

Joint squared-error nodewise regression with this penalty groups the two coefficients of each unordered edge. Its norm is nonsmooth only when the whole pair is zero, encouraging edge-wise sparsity. A fitted edge is present when its pair norm is positive. The penalty does not require equal directional coefficients, two nonzero entries in every selected group, or a symmetric positive definite precision matrix.

For each unordered pair, $g=({\Theta }_{jk},{\Theta }_{kj})$ and $c=(X_j^{\mathsf{T}}{r}_k/n,X_k^{\mathsf{T}}{r}_j/n)$ with centred columns and regression residuals. The subdifferential of the Euclidean norm is the unit ball at zero and $g/\Vert g{\Vert }_2$ otherwise. These KKT conditions are necessary and sufficient for the convex paired objective and explain joint edge deletion.