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

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.

Optimal Transport-Based Domain Adaptation for Rotated Linear Regression

Optimal Transport (OT) has proven effective for domain adaptation (DA) by aligning distributions across domains with differing statistical properties. Building on the approach of Courty et al. (2016), who mapped source data to the target domain for improved model transfer, we focus on a supervised DA problem involving linear regression models under rotational shifts. This ongoing work considers cases where source and target domains are related by a rotation-common in applications like sensor calibration or image orientation. We show that in $\mathbb{R}^2$ , when using a p-norm cost with $p $\ge$ 2$, the optimal transport map recovers the underlying rotation. Based on this, we propose an algorithm that combines K-means clustering, OT, and singular value decomposition (SVD) to estimate the rotation angle and adapt the regression model. This method is particularly effective when the target domain is sparsely sampled, leveraging abundant source data for improved generalization. Our contributions offer both theoretical and practical insights into OT-based model adaptation under geometric transformations.