  The framework of optimal transport has been leveraged to extend the notion of
rank to the multivariate setting while preserving desirable properties of the
resulting goodness-of-fit (GoF) statistics. In particular, the rank energy (RE)
and rank maximum mean discrepancy (RMMD) are distribution-free under the null,
exhibit high power in statistical testing, and are robust to outliers. In this
paper, we point to and alleviate some of the practical shortcomings of these
proposed GoF statistics, namely their high computational cost, high statistical
sample complexity, and lack of differentiability with respect to the data. We
show that all these practically important issues are addressed by considering
entropy-regularized optimal transport maps in place of the rank map, which we
refer to as the soft rank. We consequently propose two new statistics, the soft
rank energy (sRE) and soft rank maximum mean discrepancy (sRMMD), which exhibit
several desirable properties. Given $n$ sample data points, we provide
non-asymptotic convergence rates for the sample estimate of the entropic
transport map to its population version that are essentially of the order
$n^{-1/2}$ when the starting measure is subgaussian and the target measure has
compact support. This result is novel compared to existing results which
achieve a rate of $n^{-1}$ but crucially rely on both measures having compact
support. We leverage this result to demonstrate fast convergence of sample sRE
and sRMMD to their population version making them useful for high-dimensional
GoF testing. Our statistics are differentiable and amenable to popular machine
learning frameworks that rely on gradient methods. We leverage these properties
towards showcasing the utility of the proposed statistics for generative
modeling on two important problems: image generation and generating valid
knockoffs for controlled feature selection.
