  We consider support recovery in the quadratic logistic regression setting -
where the target depends on both p linear terms $x_i$ and up to $p^2$ quadratic
terms $x_i x_j$. Quadratic terms enable prediction/modeling of higher-order
effects between features and the target, but when incorporated naively may
involve solving a very large regression problem. We consider the sparse case,
where at most $s$ terms (linear or quadratic) are non-zero, and provide a new
faster algorithm. It involves (a) identifying the weak support (i.e. all
relevant variables) and (b) standard logistic regression optimization only on
these chosen variables. The first step relies on a novel insight about
correlation tests in the presence of non-linearity, and takes $O(pn)$ time for
$n$ samples - giving potentially huge computational gains over the naive
approach. Motivated by insights from the boolean case, we propose a non-linear
correlation test for non-binary finite support case that involves hashing a
variable and then correlating with the output variable. We also provide
experimental results to demonstrate the effectiveness of our methods.
