  Gromov-Wasserstein distance has found many applications in machine learning
due to its ability to compare measures across metric spaces and its invariance
to isometric transformations. However, in certain applications, this invariance
property can be too flexible, thus undesirable. Moreover, the
Gromov-Wasserstein distance solely considers pairwise sample similarities in
input datasets, disregarding the raw feature representations. We propose a new
optimal transport-based distance, called Augmented Gromov-Wasserstein, that
allows for some control over the level of rigidity to transformations. It also
incorporates feature alignments, enabling us to better leverage prior knowledge
on the input data for improved performance. We present theoretical insights
into the proposed metric. We then demonstrate its usefulness for single-cell
multi-omic alignment tasks and a transfer learning scenario in machine
learning.
