用最优传输恢复二维空间几何变换，提升小样本线性回归适应性
Geometric Domain Adaptation via Optimal Transport for Linear Regression in R^2
- 结合K均值与最优传输估计旋转、平移、缩放等几何变换
- 在p≥2的代价函数下，可精确恢复源域到目标域的变换映射
- 适用于数据稀疏场景，方法简洁可解释，适合理论研究者
最优传输近年来成为领域自适应的重要工具，通过对齐源域与目标域分布实现迁移。本文研究二维空间$ℝ^2$中源域与目标域由旋转、平移或位似关系关联的监督领域自适应问题。证明当使用$p$-范数代价且$p \geq 2$时，最优传输映射可准确恢复底层几何变换。基于此，提出一种融合K均值与最优传输的方法，用于估计变换并实现线性回归模型的适应，尤其适用于目标数据稀缺情形。模拟实验显示性能优于基线方法。不同于依赖高表达力深度学习模型，本工作聚焦经典机器学习框架，强调可解释性与理论洞察力。贡献包括建立最优传输与二维空间旋转、平移、位似变换之间的理论联系，以及一个兼具概念清晰性与应用价值的线性回归领域自适应方法。
原文摘要 · Abstract (English)
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.
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