Optimal transport (OT) distances between probability distributions are parameterized by the ground metric they use between observations. Their relevance for real-life applications strongly hinges on whether that ground metric parameter is suitably chosen. The challenge of selecting it adaptively and algorithmically from prior knowledge, the so-called ground metric learning (GML) problem, has therefore appeared in various settings. In this paper, we consider the GML problem when the learned metric is constrained to be a geodesic distance on a graph that supports the measures of interest. This imposes a rich structure for candidate metrics, but also enables far more efficient learning procedures when compared to a direct optimization over the space of all metric matrices. We use this setting to tackle an inverse problem stemming from the observation of a density evolving with time; we seek a graph ground metric such that the OT interpolation between the starting and ending densities that result from that ground metric agrees with the observed evolution. This OT dynamic framework is relevant to model natural phenomena exhibiting displacements of mass, such as the evolution of the color palette induced by the modification of lighting and materials.

概率分布之间的最佳运输（OT）距离由观测之间使用的地面度量参数化。他们对现实生活中的应用的相关性强烈地取决于该地的度量参数是否适当地选择。因此，从各种先验知识中自适应地和算法地选择它的挑战，即所谓的地面度量学习（GML）问题，已经出现在各种场合。在本文中，我们考虑了GML问题时，了解到度量约束为测地距离上一个图支持感兴趣的措施。与在所有度量矩阵空间上进行直接优化相比，这为候选度量提供了丰富的结构，但也使学习过程更加有效。我们使用此设置来解决由于观察到密度随时间变化而产生的逆问题。我们寻求一种图形地面度量，以使从该地面度量得出的开始和结束密度之间的OT插值与观测到的演变相一致。该OT动态框架与模拟表现出质量位移的自然现象有关，例如由照明和材料的修改引起的调色板的演变。