  The fixed-X knockoff filter is a flexible framework for variable selection
with false discovery rate (FDR) control in linear models with arbitrary design
matrices (of full column rank) and it allows for finite-sample selective
inference via the Lasso estimates. In this paper, we extend the theory of the
knockoff procedure to tests with composite null hypotheses, which are usually
more relevant to real-world problems. The main technical challenge lies in
handling composite nulls in tandem with dependent features from arbitrary
designs. We develop two methods for composite inference with the knockoffs,
namely, shifted ordinary least-squares (S-OLS) and feature-response product
perturbation (FRPP), building on new structural properties of test statistics
under composite nulls. We also propose two heuristic variants of S-OLS method
that outperform the celebrated Benjamini-Hochberg (BH) procedure for composite
nulls, which serves as a heuristic baseline under dependent test statistics.
Finally, we analyze the loss in FDR when the original knockoff procedure is
naively applied on composite tests.
