source: arxiv statistics ml: geometric domain adaptation via optimal transport for linear regression in r^2
level: research
optimal transport has become a key tool for domain adaptation by aligning source and target distributions. this work examines a supervised setting where domains differ by simple geometric transformations in the plane: rotation, translation, or scaling. the authors prove that when using a p-norm cost with p at least 2, the optimal transport map exactly recovers the underlying transformation. this theoretical result holds for linear regression tasks and provides a clear geometric interpretation of how optimal transport corrects distribution shifts.
based on this insight, the researchers propose a method that combines k-means clustering with optimal transport to estimate the unknown transformation. the approach first clusters the source data, then computes optimal transport between cluster centers and target samples to infer the geometric mapping. once the transformation is estimated, a linear regression model trained on source data can be adapted to the target domain. simulations show this method outperforms baseline approaches when target data is limited, without needing complex deep learning models.
the focus on classical machine learning emphasizes interpretability and theoretical understanding. by restricting to linear models and simple geometric shifts, the study explicitly characterizes how optimal transport recovers domain transformations. this contrasts with common deep domain adaptation methods that often lack clear theoretical guarantees. the findings suggest that for problems with known structural differences, simpler optimal transport-based corrections can be effective and more transparent.
why it matters: it offers a principled, interpretable way to adapt models when target data is scarce and domain shifts are geometric, useful for sensor calibration or simulation-to-real transfer.
source: arxiv statistics ml: geometric domain adaptation via optimal transport for linear regression in r^2