Lasso reduction for the Lava estimator

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.