论文标题
一个成对的多边界最佳运输系列，定义了广义度量
A Family of Pairwise Multi-Marginal Optimal Transports that Define a Generalized Metric
论文作者
论文摘要
最佳运输（OT）问题正在迅速找到机器学习的方式。偏爱其使用的是公制属性。许多问题仅对嵌入度量空间中的物体保证的解决方案接受解决方案，而非光学技术的使用会使解决方案变得复杂。多核心OT（MMOT）将OT概括为同时运输多个分布。它捕获了重要的关系，如果运输仅涉及两个分布，则会错过。但是，对MMOT的研究一直集中在其存在，独特性，实用算法和成本功能的选择上。关于MMOT的度量特性缺乏讨论，这限制了其理论和实际使用。在这里，我们证明了成对MMOT家族的新通用度量属性。我们首先解释了通过两个负面结果证明这一点的困难。之后，我们证明了MMOTS的度量属性。最后，我们表明，无法改善这个MMOT家族的广义三角不平等。我们说明了MMOT的优越性，而不是其他广义指标，而在合成和实际任务中的优越性优于非光学指标。
The Optimal transport (OT) problem is rapidly finding its way into machine learning. Favoring its use are its metric properties. Many problems admit solutions with guarantees only for objects embedded in metric spaces, and the use of non-metrics can complicate solving them. Multi-marginal OT (MMOT) generalizes OT to simultaneously transporting multiple distributions. It captures important relations that are missed if the transport only involves two distributions. Research on MMOT, however, has been focused on its existence, uniqueness, practical algorithms, and the choice of cost functions. There is a lack of discussion on the metric properties of MMOT, which limits its theoretical and practical use. Here, we prove new generalized metric properties for a family of pairwise MMOTs. We first explain the difficulty of proving this via two negative results. Afterward, we prove the MMOTs' metric properties. Finally, we show that the generalized triangle inequality of this family of MMOTs cannot be improved. We illustrate the superiority of our MMOTs over other generalized metrics, and over non-metrics in both synthetic and real tasks.
