In the context of optimal transport methods, the subspace detour approach was recently presented by Muzellec and Cuturi (2019). It consists in building a nearly optimal transport plan in the measures space from an optimal transport plan in a wisely chosen subspace, onto which the original measures are projected. The contribution of this paper is to extend this category of methods to the Gromov-Wasserstein problem, which is a particular type of transport distance involving the inner geometry of the compared distributions. After deriving the associated formalism and properties, we also discuss a specific cost for which we can show connections with the Knothe-Rosenblatt rearrangement. We finally give an experimental illustration on a shape matching problem.
翻译:在最佳运输方法方面,Muzellec和Cuturi(2019年)最近介绍了次空间绕行方法,其中包括从明智选择的子空间的最佳运输计划中建立测量空间的近乎最佳的运输计划,最初的措施将投向明智选择的子空间,本文件的贡献是将这一类方法扩大到格罗莫夫-瓦瑟斯坦问题,这是一个特殊的运输距离,涉及比较分布的内部几何学。在得出相关的形式主义和特性之后,我们还讨论了可以显示与克诺特-罗森布拉特重新排列有关联的具体成本。我们最后对形状匹配问题进行了实验性说明。