We propose a new formulation and learning strategy for computing the Wasserstein geodesic between two probability distributions in high dimensions. By applying the method of Lagrange multipliers to the dynamic formulation of the optimal transport (OT) problem, we derive a minimax problem whose saddle point is the Wasserstein geodesic. We then parametrize the functions by deep neural networks and design a sample based bidirectional learning algorithm for training. The trained networks enable sampling from the Wasserstein geodesic. As by-products, the algorithm also computes the Wasserstein distance and OT map between the marginal distributions. We demonstrate the performance of our algorithms through a series of experiments with both synthetic and realistic data.

翻译：我们提出了一个新的公式和学习战略,用于在高维的两种概率分布之间计算瓦塞斯坦大地学的大地学。通过将拉格兰梯乘数法应用于最佳运输(OT)问题的动态配方,我们得出了一个小型马克斯问题,其马鞍点是瓦塞斯坦大地学。然后我们通过深层神经网络对功能进行对称,并设计一个基于样本的双向学习算法用于培训。经过培训的网络使得能够从瓦塞斯坦大地学中取样。作为副产品,算法还计算了瓦塞尔斯坦距离和边际分布之间的奥特地图。我们通过一系列合成和现实数据实验来展示我们的算法的性能。