Whiteout: when do fixed-X knockoffs fail?

Understand

A core strength of knockoff methods is their virtually limitless customizability, allowing an analyst to exploit machine learning algorithms and domain knowledge without threatening the method's robust finite-sample false discovery rate control guarantee.

• While several previous works have investigated regimes where specific implementations of knockoffs are provably powerful, general negative results are more difficult to obtain for such a flexible method.
• In this work we recast the fixed-$X$ knockoff filter for the Gaussian linear model as a conditional post-selection inference method.
• It adds user-generated Gaussian noise to the ordinary least squares estimator $\hat\beta$ to obtain a "whitened" estimator $\widetilde\beta$ with uncorrelated entries, and performs inference using $\text{sgn}(\widetilde\beta_j)$ as the test statistic for $H_j:\; \beta_j = 0$.