RANK: Large-Scale Inference With Graphical Nonlinear Knockoffs

Abstract

Power and reproducibility are key to enabling refined scientific discoveries in contemporary big data applications with general high-dimensional nonlinear models. In this article, we provide theoretical foundations on the power and robustness for the model-X knockoffs procedure introduced recently in Candès, Fan, Janson and Lv in high-dimensional setting when the covariate distribution is characterized by Gaussian graphical model. We establish that under mild regularity conditions, the power of the oracle knockoffs procedure with known covariate distribution in high-dimensional linear models is asymptotically one as sample size goes to infinity. When moving away from the ideal case, we suggest the modified model-X knockoffs method called graphical nonlinear knockoffs (RANK) to accommodate the unknown covariate distribution. We provide theoretical justifications on the robustness of our modified procedure by showing that the false discovery rate (FDR) is asymptotically controlled at the target level and the power is asymptotically one with the estimated covariate distribution. To the best of our knowledge, this is the first formal theoretical result on the power for the knockoffs procedure. Simulation results demonstrate that compared to existing approaches, our method performs competitively in both FDR control and power. A real dataset is analyzed to further assess the performance of the suggested knockoffs procedure. Supplementary materials for this article are available online.

Keywords:

• Big data
• Graphical nonlinear knockoffs
• High-dimensional nonlinear models
• Large-scale inference and FDR
• Power
• Reproducibility
• Robustness

Acknowledgment

The authors sincerely thank the Joint Editor, Associate Editor, and referees for their valuable comments that helped improve the article substantially.

Additional information

Funding

This work was supported by NIH Grant 1R01GM131407-01, NSF CAREER Award DMS-1150318, a grant from the Simons Foundation, and Adobe Data Science Research Award. Gaorong Li’s research was supported by the National Natural Science Foundation of China (grant number 11871001) and Beijing Natural Science Foundation (grant number 1182003).