Computation and Consistent Estimation of Stationary Optimal Transport Plans

Informally, the optimal transport (OT) problem is to align, or couple, two distributions of interest as best as possible with respect to some prespecified cost. A coupling that achieves the minimum cost among all couplings is referred to as an OT plan; the cost of the OT plan is referred to as the OT cost. Researchers in statistics and machine learning have expended a great deal of effort to understand the properties of OT plans and costs. The motivation for this work stems partly from the fact that, unlike many other divergence measures and metrics between distributions, OT plans and costs describe relationships between distributions in a manner that respects the geometry of the underlying space (by way of the specified cost). However, this advantage does not necessarily carry over when standard OT techniques are applied to distributions with specific structure. In the case that the two distributions describe stationary stochastic processes, the OT problem may ignore the differences in the sequential dependence of either process. One must find a way to make the OT problem account for the stationary dependence of the marginal processes. In this thesis, we study OT for stationary processes, a field that we refer to as stationary optimal transport. Through example and theory, we argue that when applying OT to stationary processes, one should incorporate the stationarity into the problem directly -- constraining the set of allowed transport plans to those that are stationary themselves. In this way, we only consider transport plans that respect the dependence structure of the marginal processes. We study this constrained OT problem from statistical and computational perspectives, with an eye toward applications in machine learning and data science. In particular, we develop algorithms for computing stationary OT plans of Markov chains, extend these tools for Markov OT to the alignment and comparison of weighted graphs, and propose estimates of stationary OT plans based on finite sequences of observations. We build upon existing techniques in OT as well as draw from a variety of fields including Markov decision processes, graph theory, and ergodic theory. In doing this, we uncover new perspectives on OT and pave the way for additional applications and approaches in future work.Doctor of Philosoph