  Optimal transport (OT) is a powerful geometric tool used to compare and align
probability measures following the least effort principle. Despite its
widespread use in machine learning (ML), OT problem still bears its
computational burden, while at the same time suffering from the curse of
dimensionality for measures supported on general high-dimensional spaces. In
this paper, we propose to tackle these challenges using representation
learning. In particular, we seek to learn an embedding space such that the
samples of the two input measures become alignable in it with a simple affine
mapping that can be calculated efficiently in closed-form. We then show that
such approach leads to results that are comparable to solving the original OT
problem when applied to the transfer learning task on which many OT baselines
where previously evaluated in both homogeneous and heterogeneous DA settings.
The code for our contribution is available at
\url{https://github.com/Oleffa/LaOT}.
