Optimal Transport-Based Domain Adaptation for Rotated Linear Regression

TL;DR

This paper introduces a novel optimal transport-based method for domain adaptation in linear regression tasks involving rotational shifts, combining clustering, OT, and SVD to accurately estimate rotations and improve model transfer, especially with sparse target data.

Contribution

It presents a new algorithm that leverages OT and SVD to recover rotations in domain adaptation, providing theoretical and practical advancements for geometric transformations.

Findings

The OT map recovers the underlying rotation in 2D with p-norm cost p ≥ 2.

The proposed method effectively estimates rotation angles using clustering, OT, and SVD.

The approach improves regression model transfer, especially with sparse target domain samples.

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.