Many numerical algorithms and learning tasks rest on solution of the Monge-Kantorovich problem and corresponding Wasserstein distances. While the natural approach is to treat the problem as an infinite-dimensional linear programming, such a methodology severely limits the computational performance due to the polynomial scaling with respect to the sample size along with intensive memory requirements. We propose a novel alternative framework to address the Monge-Kantorovich problem based on a projection type gradient descent scheme. The micro-dynamics is built on the notion of the conditional expectation, where the connection with the opinion dynamics is explored and leveraged to build compact numerical schemes. We demonstrate that the devised dynamics recovers random maps with favourable computational performance. Along with the theoretical insight, the provided dynamics paves the way for innovative approaches to construct numerical schemes for computing optimal transport maps as well as Wasserstein distances.

本研究针对传统的Monge-Kantorovich问题所面临的计算性能瓶颈，提出了一种基于投影梯度下降的新框架。该方法通过微观动力学与条件期望的联系，展示了在计算最优运输映射和Wasserstein距离方面的显著优势，具有较好的计算性能和创新的数值方案构建潜力。

正交耦合动力学下的最优运输