Compatibility constant (${\varphi }_{\Sigma }$)

For a positive-semidefinite matrix $\Sigma$, an active set $S$ of size $s$, and cone factor three, one compatibility constant is defined byIt measures how strongly the quadratic form controls active-coordinate ${\ell }^1$ error on the Lasso cone condition.

If $\Vert \Theta -\Sigma {\Vert }_{\max }\le {\varphi }_{\Sigma }^2/(32s)$, then ${\varphi }_{\Theta }^2\ge {\varphi }_{\Sigma }^2/2$. Indeed, every normalized cone vector has $\Vert \delta {\Vert }_1\le 4$, so its quadratic form changes by at most $16\Vert \Theta -\Sigma {\Vert }_{\max }\le {\varphi }_{\Sigma }^2/(2s)$.