  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, has a complexity scaling as $O(p^3)$
where $p$ is the ambient dimension, while another assumes a rank $k$ factor
model on the covariance matrix to reduce this complexity bound to $O(pk^2)$. We
also derive efficient procedures to both estimate factor models and sample
knockoff covariates with complexity linear in the dimension. We test our
methods on problems with $p$ as large as $500,000$.
