  In a standard regression problem, we have a set of explanatory variables
whose effect on some response vector is modeled. For wide binary data, such as
genetic marker data, we often have two limitations. First, we have more
parameters than observations. Second, main effects are not the main focus;
instead the primary aim is to uncover interactions between the binary variables
that effect the response. Methods such as logic regression are able to find
combinations of the explanatory variables that capture higher-order
relationships in the response. However, the number of explanatory variables
these methods can handle is highly limited. To address these two limitations we
need to reduce the number of variables prior to computationally demanding
analyses. In this paper, we demonstrate the usefulness of using so-called
cross-leverage scores as a means of sampling subsets of explanatory variables
while retaining the valuable interactions.
