On the multi-marginal optimal partial transport and partial barycenter problems

Abstract

Relatively recently, there has been much activity on two particular generalizations of the classical two-marginal optimal transport problem. The first is the partial transport problem, where the total mass of the two distributions to be coupled may not match, and one is forced to choose submeasures of the constraints for coupling. The other generalization is the multi-marginal transport problem, where there are 3 or more probability distributions to be coupled together in some optimal manner. By combining the above two generalizations we obtain a natural extension: the multi-marginal optimal partial transport problem. In joint work with Brendan Pass (University of Alberta), we have obtained uniqueness of solutions (under hypotheses analogous to the two-marginal partial transport problem given by Figalli) by relating the problem to what we deem the “partial barycenter problem” for finite measures. Interestingly enough, we also observe some significantly different behavior of solutions compared to the two marginal case.