Ground Metric Learning on Graphs
Ground Metric Learning on Graphs
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DOI：
10.1007/s10851-020-00996-z
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发表时间：
2020-10-30
影响因子：
2
通讯作者：
Peyre, Gabriel
中科院分区：
文献类型：
--
作者：
Heitz, Matthieu;Bonneel, Nicolas;Peyre, Gabriel
概率分布之间的最优运输（OT）距离由它们在观测之间使用的地面度量参数化。它们与实际应用的相关性在很大程度上取决于是否适当地选择了地面度量参数。因此，从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，已经出现在各种环境中。在本文中，我们考虑的GML问题时，学习的度量被约束为一个图上的测地线距离，支持感兴趣的措施。这为候选度量强加了丰富的结构，但与在所有度量矩阵的空间上的直接优化相比，也实现了更有效的学习过程。我们使用这种设置来解决一个逆问题，从观察到的密度随时间的变化，我们寻求一个图形地面度量，使OT插值的开始和结束的密度，从地面度量的结果与观察到的演变。这个OT动态框架与展示质量位移的模型自然现象有关，例如由照明和材料的修改引起的调色板的演变。
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.