Optimal Transport-Based Domain Adaptation for Rotated Linear Regression

2025-05-14

AI Summary

This paper addresses supervised domain adaptation under rotational shift between source and target domains, focusing on improving the cross-domain generalization of linear regression models. We first provide the first theoretical proof that the optimal transport (OT) mapping under the ℓₚ-norm (p ≥ 2) in ℝ² exactly recovers rotational transformations. Building on this insight, we propose a three-stage adaptive framework integrating K-means clustering, OT-based distribution alignment, and SVD-based rotation estimation. The method operates without target-label supervision and requires only a small number of unlabeled target samples to robustly estimate and correct for rotational misalignment. Experiments on rotation-sensitive tasks—including sensor calibration and image orientation estimation—demonstrate substantial improvements in regression accuracy. Our approach thus bridges rigorous theoretical guarantees with practical efficacy in rotational domain adaptation.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Computer Vision: Adversarial Attacks & Robustness
• Application Domains: Transportation

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

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.

Problem

• Aligning distributions across domains with rotational shifts
• Estimating rotation angle for linear regression adaptation
• Improving model transfer with sparse target domain data

Innovation

• Optimal Transport aligns domains with rotational shifts
• Combines K-means, OT, and SVD for rotation estimation
• Leverages source data to improve sparse target generalization