Learning generative models for valid knockoffs using novel multivariate-rank based statistics

We consider the problem of generating valid knockoffs for knockoff filtering which is a statistical method that provides provable false discovery rate guarantees for any model selection procedure. To this end, we are motivated by recent advances in multivariate distribution-free goodness-of-fit tests namely, the rank energy (RE), that is derived using theoretical results characterizing the optimal maps in the Monge's Optimal Transport (OT) problem. However, direct use of use RE for learning generative models is not feasible because of its high computational and sample complexity, saturation under large support discrepancy between distributions, and non-differentiability in generative parameters. To alleviate these, we begin by proposing a variant of the RE, dubbed as soft rank energy (sRE), and its kernel variant called as soft rank maximum mean discrepancy (sRMMD) using entropic regularization of Monge's OT problem. We then use sRMMD to generate deep knockoffs and show via extensive evaluation that it is a novel and effective method to produce valid knockoffs, achieving comparable, or in some cases improved tradeoffs between detection power Vs false discoveries.