论文标题
桥接度量嵌入和最佳运输的理论保证
Theoretical Guarantees for Bridging Metric Measure Embedding and Optimal Transport
论文作者
论文摘要
我们提出了一种新颖的方法，用于比较支持不一定位于相同度量空间的分布。与Gromov-Wasserstein（GW）的距离（GW）比较了每个分布中元素的成对距离，我们考虑了一种方法，允许将度量测量空间嵌入共同的欧几里得空间中，并在嵌入式分布上计算最佳运输（OT）。这导致了我们所谓的subderding强大的Wasserstein（SERW）距离。在某些条件下，SERW是一种考虑使用公共度量的（低延伸）嵌入式分布的OT距离的距离。除了概括了最近几项OT作品的这一新颖的建议外，我们的贡献还基于几个理论分析：（i）我们表征了嵌入空间以定义SERW距离以进行分配对准； （ii）我们证明SERW模仿GW距离的属性几乎相同，并且我们在GW和Serw之间提供了成本关系。该论文还提供了一些数字插图，介绍了SERW在匹配问题上的行为。
We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we consider a method allowing to embed the metric measure spaces in a common Euclidean space and compute an optimal transport (OT) on the embedded distributions. This leads to what we call a sub-embedding robust Wasserstein (SERW) distance. Under some conditions, SERW is a distance that considers an OT distance of the (low-distorted) embedded distributions using a common metric. In addition to this novel proposal that generalizes several recent OT works, our contributions stand on several theoretical analyses: (i) we characterize the embedding spaces to define SERW distance for distribution alignment; (ii) we prove that SERW mimics almost the same properties of GW distance, and we give a cost relation between GW and SERW. The paper also provides some numerical illustrations of how SERW behaves on matching problems.
