A new pairwise cost function is proposed for the optimal transport barycenter problem, adopting the form of the minimal action between two points, with a Lagrangian that takes into account an underlying probability distribution. Under this notion of distance, two points can only be close if there exist paths joining them that do not traverse areas of small probability. A framework is proposed and developed for the numerical solution of the corresponding data-driven optimal transport problem. The procedure parameterizes the paths of minimal action through path dependent Chebyshev polynomials and enforces the agreement between the paths' endpoints and the given source and target distributions through an adversarial penalization. The methodology and its application to clustering and matching problems is illustrated through synthetic examples.

翻译：本文针对最优传输重心问题提出了一种新的成对成本函数，该函数采用两点间最小作用量的形式，其拉格朗日量考虑了基础概率分布。在这种距离概念下，两点仅当存在连接它们的路径且该路径不穿过低概率区域时才能被视为接近。我们提出并发展了一个用于数值求解相应数据驱动最优传输问题的框架。该框架通过路径依赖的切比雪夫多项式参数化最小作用量路径，并通过对抗性惩罚确保路径端点与给定源分布及目标分布的一致性。通过合成示例，我们展示了该方法论及其在聚类和匹配问题中的应用。