Supplement to “ Covariate Assisted Screening and Estimation ”

As mentioned before, the success of CASE relies on two noteworthy properties: the Sure Screening (SS) property and the Separable After Screening (SAS) property. In this section, we discuss the two properties in detail, and illustrate how these properties enable us to decompose the original regression problem to many small-size regression problems which can be fit separately. We then use these properties to prove Theorem 2.2. We start with the SS property. Recall that U∗ p is the set of all retained indices at the end of the PS-step. The following lemma is proved in Section B.