Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport
Organizations: BIFOLD – Berlin Institute for the Foundations of Learning and Data, 10587 Berlin, Germany · Machine Learning Group, Technische Universität Berlin, 10587 Berlin, Germany · Department of Artificial Intelligence, Korea University, Anam-dong, Seongbuk-gu, Seoul 02841, Korea · Max Planck Institute for Informatics, Stuhlsatzenhausweg, 66123 Saarbrücken, Germany · Institute for AI in Medicine, Charité – Universitätsmedizin Berlin, 10117 Berlin, Germany
Abstract
Optimal transport (OT) is a central framework for modeling distribution shifts. Because OT compares distributions directly in input space, a well-designed ground metric between observations is essential to ensure that the optimizer does not violate the true geometry of change. We propose Displacement-Reshaped Optimal Transport (ReshapeOT), a method that reshapes the ground metric by integrating observed sample displacements as an additional source of knowledge. Technically, ReshapeOT replaces the Euclidean metric with a Mahalanobis distance estimated from displacement second moments. This effectively carves expressways through the input space, inviting transport solutions that better align with observed displacements. Our method is computationally lightweight, integrates seamlessly into any OT solver that operates on a cost matrix, and can be kernelized for further flexibility. Experiments on synthetic and real-world data show that ReshapeOT achieves substantial gains in transport reliability. We further demonstrate our method's usefulness in two practical use cases.