Non-Aligned Distribution Distance using Metric Measure Embedding and Optimal Transport

We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance that compares pairwise distance of elements from each distribution, we consider a method that embeds the metric measure spaces in a common Euclidean space and computes an optimal transport (OT) on the embedded distributions. This leads to what we call a sub-embedding robust Wasserstein(SERW). 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 experiments illustrating how SERW behaves on matching problems in real-world.