Estimation of stationary optimal transport plans

Abstract

Abstract We study optimal transport for stationary stochastic processes taking values in finite spaces. In order to reflect the stationarity of the underlying processes, we restrict attention to stationary couplings, also known as joinings. The resulting optimal joining problem captures differences in the long-run average behavior of the processes of interest. We introduce estimators of both optimal joinings and the optimal joining cost, and establish consistency of the estimators under mild conditions. Furthermore, under stronger mixing assumptions we establish finite-sample error rates for the estimated optimal joining cost that extend the best known results in the iid case. We also extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem. Finally, we validate our convergence results empirically as well as demonstrate the computational advantage of the entropic problem in a simulation experiment.