Approximation of Optimal Transport problems with marginal moments constraints
This work offers a practical approximation method for OT problems that mitigates the curse of dimensionality, benefiting researchers and practitioners in fields like economics and physics who need scalable OT solutions.
The authors propose relaxing optimal transport (OT) problems by replacing marginal constraints with moment constraints, proving that the resulting problem is solved by a finite discrete measure with linear scaling in the number of marginals. They establish convergence rates of O(1/n) or O(1/n^2) in the number of moments and provide algorithms with numerical examples.
Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. This approximation method is also relevant for Martingale OT problems. We show the convergence of the MCOT problem toward the corresponding OT problem. In some fundamental cases, we obtain rates of convergence in $O(1/n)$ or $O(1/n^2)$ where $n$ is the number of moments, which illustrates the role of the moment functions. Last, we present algorithms exploiting the fact that the MCOT is reached by a finite discrete measure and provide numerical examples of approximations.