用最优传输方法解决旋转偏移下的线性回归域适应问题
Optimal Transport-Based Domain Adaptation for Rotated Linear Regression
- 结合K-means、OT与SVD，估计旋转角度并适配模型
- 在二维空间中，当p≥2时能精确恢复旋转关系
- 特别适用于目标域数据稀疏的场景
最优传输（OT）在域适应（DA）中表现出色，可通过对齐不同统计分布来提升模型迁移效果。本文基于Courty等（2016）的方法，聚焦于受旋转偏移影响的监督线性回归域适应问题，此类偏移常见于传感器校准或图像方向识别。研究发现，在ℝ²中使用p范数代价且p≥2时，最优传输映射可准确恢复原始旋转。据此提出一种结合K-means聚类、最优传输与奇异值分解（SVD）的算法，用于估计旋转角并实现模型适配。该方法在目标域样本稀疏时尤为有效，能充分利用源域丰富数据提升泛化能力。研究成果为几何变换下的OT模型适配提供了理论与实践双重启示。
原文摘要 · Abstract (English)
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.
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