# Variable Selection with the Knockoffs: Composite Null Hypotheses

arXiv: https://arxiv.org/abs/2203.02849 · HTML: https://mlsanity.com/paper/2203.02849/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 23 references that are in the map.
- First posted on arXiv 6 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [Controlling the false discovery rate via knockoffs](https://mlsanity.com/paper/1404.5609/) (2014)
- [DeepPINK: reproducible feature selection in deep neural networks](https://mlsanity.com/paper/1809.01185/) (2018)
- [Deep Knockoffs](https://mlsanity.com/paper/1811.06687/) (2018)
- [A knockoff filter for high-dimensional selective inference](https://mlsanity.com/paper/1602.03574/) (2016)
- [Robust inference with knockoffs](https://mlsanity.com/paper/1801.03896/) (2018)
- [Conditional calibration for false discovery rate control under dependence](https://mlsanity.com/paper/2007.10438/) (2020)
- [Powerful Knockoffs via Minimizing Reconstructability](https://mlsanity.com/paper/2011.14625/) (2020)
- [IPAD: Stable Interpretable Forecasting with Knockoffs Inference](https://mlsanity.com/paper/1809.05032/) (2018)
- [Differentially Private Variable Selection via the Knockoff Filter](https://mlsanity.com/paper/2109.05402/) (2021)

## Abstract (arXiv)

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.