Empirical Regularized Optimal Transport: Statistical Theory and Applications

We derive limit distributions for various empirical regularized optimal transport quantities between probability distributions supported on a finite metric space and show their bootstrap consistency. In particular, we prove that the empirical regularized transport plan itself asymptotically follows a Gaussian law. The theory includes the Boltzmann--Shannon entropy regularization and hence a limit law for the widely applied Sinkhorn divergence. Our approach is based on parametric optimization techniques for the regularized transport problem in conjunction with a statistical delta method. The asymptotic results are investigated in Monte Carlo simulations. We further discuss computational consequences and statistical applications, e.g., confidence bands for colocalization analysis of protein interaction networks based on regularized optimal transport.