FANOK: Knockoffs in Linear Time

Abstract

We describe a series of algorithms that efficiently implement Gaussian model-X knockoffs to control the false discovery rate on large-scale feature selection problems. Identifying the knockoff distribution requires solving a large-scale semidefinite program for which we derive several efficient methods. One handles generic covariance matrices and has a complexity scaling as $\mathcal{O}(p^3)$, where $p$ is the ambient dimension, while another assumes a rank-$k$ factor model on the covariance matrix to reduce this complexity bound to $\mathcal{O}(pk^2)$. We review an efficient procedure to estimate factor models and show that under a factor model assumption, we can sample knockoff covariates with complexity linear in the dimension. We test our methods on problems with $p$ as large as 500 000.