Geometric Domain Adaptation via Optimal Transport for Linear Regression in R^2

2026-06-11

AI Summary

This work addresses the challenge of domain adaptation when target-domain data are scarce and geometric transformations—such as rotation, translation, or scaling—between source and target domains render conventional linear regression-based transfer ineffective. The authors propose a novel domain adaptation method that integrates K-means clustering with optimal transport. Theoretically, they establish—for the first time—that under p≥2 norm cost functions, the optimal transport map exactly recovers rigid and similarity transformations in two-dimensional space. Methodologically, the approach operates without deep learning architectures, instead estimating an interpretable underlying geometric mapping from the source to the target domain to enable effective model transfer. Empirical results demonstrate that the proposed method significantly outperforms existing baselines in low-data target scenarios, offering both theoretical rigor and practical utility.

Abstract

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.

Problem

• domain adaptation
• optimal transport
• linear regression
• geometric transformation
• R^2

Innovation

• Optimal Transport
• Geometric Domain Adaptation
• Linear Regression
• Homothety
• K-means