Optimal Transportation by Orthogonal Coupling Dynamics

Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an infinite-dimensional linear programming, such a methodology 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 dynamics builds on the notion of the conditional expectation, where the connection with the opinion dynamics is leveraged to devise efficient numerical schemes. We demonstrate that the resulting dynamics recovers random maps with favourable computational performance. Along with the theoretical insight, the proposed dynamics paves the way for innovative approaches to construct numerical schemes for computing optimal transport maps as well as Wasserstein distances.