  Classical optimal transport problem seeks a transportation map that preserves
the total mass betwenn two probability distributions, requiring their mass to
be the same. This may be too restrictive in certain applications such as color
or shape matching, since the distributions may have arbitrary masses and/or
that only a fraction of the total mass has to be transported. Several
algorithms have been devised for computing partial Wasserstein metrics that
rely on an entropic regularization, but when it comes with exact solutions,
almost no partial formulation of neither Wasserstein nor Gromov-Wasserstein are
available yet. This precludes from working with distributions that do not lie
in the same metric space or when invariance to rotation or translation is
needed. In this paper, we address the partial Wasserstein and
Gromov-Wasserstein problems and propose exact algorithms to solve them. We
showcase the new formulation in a positive-unlabeled (PU) learning application.
To the best of our knowledge, this is the first application of optimal
transport in this context and we first highlight that partial Wasserstein-based
metrics prove effective in usual PU learning settings. We then demonstrate that
partial Gromov-Wasserstein metrics is efficient in scenario where point clouds
come from different domains or have different features.
