Whiteout: when do fixed-X knockoffs fail?
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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 β̂ to obtain a "whitened" estimator β with uncorrelated entries, and performs inference using sgn(β_j) as the test statistic for H_j: β_j = 0. We prove equivalence between our whitening formulation and the more standard formulation involving negative control predictor variables, showing how the fixed-X knockoffs framework can be used for multiple testing on any problem with (asymptotically) multivariate Gaussian parameter estimates. Relying on this perspective, we obtain the first negative results that universally upper-bound the power of all fixed-X knockoff methods, without regard to choices made by the analyst. Our results show roughly that, if the leading eigenvalues of Var(β̂) are large with dense leading eigenvectors, then there is no way to whiten β̂ without irreparably erasing nearly all of the signal, rendering sgn(β_j) too uninformative for accurate inference. We give conditions under which the true positive rate (TPR) for any fixed-X knockoff method must converge to zero even while the TPR of Bonferroni-corrected multiple testing tends to one, and we explore several examples illustrating this phenomenon.
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