The active-set constraints for the Lasso path

Let ${\mathcal A}$ be the active set chosen by the Lasso, and let ${\mathcal A}_0$, ${\mathcal A}_1$, and ${\mathcal A}_{10}$ be the sets defined in the source. Let $A_{\mathcal A}$, ${\mathcal G}_{\mathcal A}$, $w_{\mathcal A}$, and $\widehat c_j$ denote the corresponding quantities, with

$$w_{\mathcal A}=A_{\mathcal A}{\mathcal G}_{\mathcal A}^{-1}{\bf 1}_{\mathcal A}.$$

Lasso active-set constraints. The Lasso choice must satisfy ${\mathcal A}_1\subseteq {\mathcal A}$ and ${\mathcal A}\subseteq {\mathcal A}_{10}$; moreover, $w_{\mathcal A}$ cannot have $\operatorname{sign}(w_j)\ne\operatorname{sign}(\widehat c_j)$ for any $j\in{\mathcal A}_0$, and, subject to these first three constraints, ${\mathcal A}$ must minimize $A_{\mathcal A}$.

These are consequences of the preceding lemmas and characterize the constraints used to compare the Lasso and LARS--Lasso paths. The source presents them as consequences rather than as an independently named conjecture; their resolution is therefore not separately established here.