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Optimal Transport-Based Domain Adaptation for Rotated Linear Regression

Brian Britos, Mathias Bourel

TL;DR

This work tackles domain adaptation for supervised linear regression under unknown rotational shifts between source and target domains. It leverages Optimal Transport to align distributions and proves in $\mathbb{R}^2$ that, with a $p$-norm cost and $p \ge 2$, the OT map recovers the underlying rotation, enabling rotation-angle estimation via a K-means–OT–SVD pipeline. The method is extended to higher dimensions by projecting data onto a 2D plane with PCA before applying the rotated-regression procedure. Empirically, the approach improves median predictive accuracy on sparsely sampled target domains when abundant source data is available, offering a practical strategy for settings like sensor calibration and image orientation under geometric transformations.

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.

Optimal Transport-Based Domain Adaptation for Rotated Linear Regression

TL;DR

This work tackles domain adaptation for supervised linear regression under unknown rotational shifts between source and target domains. It leverages Optimal Transport to align distributions and proves in that, with a -norm cost and , the OT map recovers the underlying rotation, enabling rotation-angle estimation via a K-means–OT–SVD pipeline. The method is extended to higher dimensions by projecting data onto a 2D plane with PCA before applying the rotated-regression procedure. Empirically, the approach improves median predictive accuracy on sparsely sampled target domains when abundant source data is available, offering a practical strategy for settings like sensor calibration and image orientation under geometric transformations.

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 , when using a p-norm cost with \ge, 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.
Paper Structure (6 sections, 1 equation, 3 figures, 2 algorithms)

This paper contains 6 sections, 1 equation, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our method
  3. Principal theoretical result
  4. Simulations
  5. Varying the cardinality of $\mathcal{D}^s$
  6. Varying the angle $\theta$ and noise variance $\sigma$

Figures (3)

  • Figure 1: Red points are the source set, blue ones the target set and the purple dots are the $K$-means centroids. Step 0 is the raw data, in Step 1 we apply $K$-means with $K = n_t$ and in Step 3 we find the optimal transport plan.
  • Figure 2: Comparison of the performance of our method against the standard linear regression across different settings of $n_s$.
  • Figure 3: Median variation obtained varying the angle rotation $\theta$ and noise level $\sigma$.