  Consider a linear model $y = X \beta + z$, $z \sim N(0, \sigma^2 I_n)$. The
Gram matrix $\Theta = \frac{1}{n} X'X$ is non-sparse, but it is approximately
the sum of two components, a low-rank matrix and a sparse matrix, where neither
component is known to us. We are interested in the Rare/Weak signal setting
where all but a small fraction of the entries of $\beta$ are nonzero, and the
nonzero entries are relatively small individually. The goal is to rank the
variables in a way so as to maximize the area under the ROC curve.
  We propose Factor-adjusted Covariate Assisted Ranking (FA-CAR) as a two-step
approach to variable ranking. In the FA-step, we use PCA to reduce the linear
model to a new one where the Gram matrix is approximately sparse. In the
CAR-step, we rank variables by exploiting the local covariate structures.
  FA-CAR is easy to use and computationally fast, and it is effective in
resolving signal cancellation, a challenge we face in regression models. FA-CAR
is related to the recent idea of Covariate Assisted Screening and Estimation
(CASE), but two methods are for different goals and are thus very different.
  We compare the ROC curve of FA-CAR with some other ranking ideas on numerical
experiments, and show that FA-CAR has several advantages. Using a Rare/Weak
signal model, we derive the convergence rate of the minimum sure-screening
model size of FA-CAR. Our theoretical analysis contains several new
ingredients, especially a new perturbation bound for PCA.
