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

Learning representations that are closed-form Monge mapping optimal with application to domain adaptation