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