Optimal Transport has become recently a powerful method for domain adaptation by aligning source and target distributions. We study a supervised domain adaptation problem where source and target domains are related by a rotation or a translation or a homothety in $\mathbb{R}^2$. We prove that the optimal transport map recovers the underlying map when using a $p-$norm cost with $p \geq 2$. Based on this insight, we develop a method combining $K-$means and optimal transport to estimate the underlying map, enabling adaptation of linear regression models when target data is scarce. Simulations demonstrate improved performance over baseline methods. Rather than relying on highly expressive deep learning architectures, we focus on classical machine learning models to emphasize interpretability and theoretical insight. This perspective allows us to explicitly characterize the role of optimal transport in recovering geometric transformations such as rotations, translations, and homotheties. Our contributions include a theoretical result linking optimal transport and rotations, translations and homothecies in $\mathbb{R}^2$, and a practical method for adaptation in linear regression offering both conceptual clarity and applied value in domain adaptation tasks in this space.

翻译：最优传输通过对齐源域和目标域分布，近年已成为域自适应的强有力方法。本文研究一类有监督域自适应问题，其中源域与目标域在$\mathbb{R}^2$中通过旋转、平移或齐次变换相关联。我们证明，当采用$p\geq 2$的$p-$范数代价函数时，最优传输映射能够恢复潜在的几何变换。基于这一发现，我们提出一种结合$K-$均值与最优传输的方法来估计潜在映射，从而在目标数据稀缺时实现线性回归模型的自适应。仿真结果表明该方法优于基线模型。本研究不依赖高表达能力的深度学习架构，而是聚焦于经典机器学习模型以强调可解释性与理论洞见。这一视角使我们能够明确刻画最优传输在恢复旋转、平移和齐次变换等几何变换中的作用。本文贡献包括：在$\mathbb{R}^2$中建立最优传输与旋转、平移及齐次变换之间关联的理论结果，以及一种兼具概念清晰性与实用价值的线性回归自适应方法，适用于该空间中的域自适应任务。