Power analysis of knockoff filters for correlated designs

The knockoff filter introduced by Barber and Cand\ es 2016 is an elegant framework for controlling the false discovery rate in variable selection. While empirical results indicate that this methodology is not too conservative, there is no conclusive theoretical result on its power. When the predictors are i.i.d.\ Gaussian, it is known that as the signal to noise ratio tend to infinity, the knockoff filter is consistent in the sense that one can make FDR go to 0 and power go to 1 simultaneously. In this work we study the case where the predictors have a general covariance matrix \bsigma . We introduce a simple functional called \emph{effective signal deficiency (ESD)} of the covariance matrix of the predictors that predicts consistency of various variable selection methods.