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}.

翻译：最优传输（OT）是一种强大的几何工具，用于遵循最小努力原则比较和对齐概率测度。尽管其在机器学习（ML）中广泛使用，OT问题仍存在计算负担，同时对于支撑在一般高维空间上的测度还受维度灾难困扰。本文提出利用表示学习应对这些挑战。具体而言，我们致力于学习一个嵌入空间，使得两个输入测度的样本在该空间中可通过简单的仿射映射对齐，且该映射能高效闭式计算。我们进而证明，当该方法应用于迁移学习任务（众多OT基线方法此前已在同质与异质领域自适应设置下进行过评估）时，其结果可与求解原始OT问题相媲美。相关代码已开源在\url{https://github.com/Oleffa/LaOT}。