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.

中文翻译：

---

具有边际矩约束的最优运输问题的近似

最优传输 (OT) 问题出现在从物理学到经济学的广泛应用中。获得这些问题的数值近似解是一个具有实际重要性的具有挑战性的问题。在这项工作中，我们研究了当边际约束被一些矩约束取代时 OT 问题的松弛。使用 Tchakaloff 定理，我们表明矩约束最优传输问题 (MCOT) 是通过有限离散度量实现的。有趣的是，对于多边际 OT 问题，由该度量加权的点数与边际定律的数量成线性比例，这鼓励绕过维度诅咒。这种近似方法也与 Martingale OT 问题相关。我们展示了 MCOT 问题对相应 OT 问题的收敛性。在一些基本情况下，我们在 $O(1/n)$ 或 $O(1/n^2)$ 中获得收敛率，其中 $n$ 是矩的数量，这说明了矩函数的作用。最后，我们提出了利用 MCOT 通过有限离散度量达到这一事实的算法，并提供了近似值的数值示例。