  This paper introduces a machine for sampling approximate model-X knockoffs
for arbitrary and unspecified data distributions using deep generative models.
The main idea is to iteratively refine a knockoff sampling mechanism until a
criterion measuring the validity of the produced knockoffs is optimized; this
criterion is inspired by the popular maximum mean discrepancy in machine
learning and can be thought of as measuring the distance to pairwise
exchangeability between original and knockoff features. By building upon the
existing model-X framework, we thus obtain a flexible and model-free
statistical tool to perform controlled variable selection. Extensive numerical
experiments and quantitative tests confirm the generality, effectiveness, and
power of our deep knockoff machines. Finally, we apply this new method to a
real study of mutations linked to changes in drug resistance in the human
immunodeficiency virus.
