Estimation of Stationary Optimal Transport Plans

TL;DR

This paper investigates stationary optimal transport for stochastic processes, introducing estimators for optimal joinings and costs, proving their consistency, and providing finite-sample error rates under mixing conditions.

Contribution

It develops consistent estimators for stationary optimal transport plans and costs, extending finite-sample error bounds to dependent processes with entropy regularization.

Findings

Establishes consistency of estimators for stationary optimal joinings.

Provides finite-sample error rates under mixing conditions.

Extends analysis to entropy-penalized optimal joinings.

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 we 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. Finally, we extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem.