Lava estimator ($\hat{\delta }+\hat{\beta }$)

The Lava estimator minimizes $\Vert Y-X(\delta +\beta ){\Vert }_2^2/(2n)+{\lambda }_1\Vert \delta {\Vert }_1+{\lambda }_2\Vert \beta {\Vert }_2^2$. It combines a component selected by Lasso with a component shrunk by ridge regression. Profiling the latter yields a transformed Lasso problem; this is not the same coefficient penalty as elastic net regularization.

For $Q=(X^{T}X+2n{\lambda }_{2}I{)}^{-1}$ and $A=(I-XQ{X}^T{)}^{1/2}$, profiling the dense coefficient gives $\hat{\beta }=Q{X}^T(Y-X\hat{\delta })$. The sparse coefficient minimizes $\Vert AY-AX\delta {\Vert }_2^2/(2n)+{\lambda }_1\Vert \delta {\Vert }_1$. The transformation uses the principal square root of a positive semidefinite matrix and remains valid for rank-deficient designs.