Learning Transferable Representations Across Distributional Shifts via Koopman Operator and Optimal Transport

In this paper, we propose a domain adaptation method that learns cross domain representations by connecting optimal transport and transfer operators theory. We model the source and target distributions as Gaussian mixtures and solve an entropic optimal transport problem between their components, providing a discrete backbone that captures cross-domain correspondences. We then leverage the component densities to lift this backbone into a continuous artificial dynamics designed to preserve semantic class structure, and characterize this dynamics through its Koopman operator. We motivate how, under favorable connectivity conditions, the slow eigenmodes provide a representation that is stable across domains and naturally defined throughout the ambient space, thereby enabling out-of-sample mapping. Moreover, our operator construction confines the dynamics to an explicit finite-dimensional function space, so that the representation is structurally prescribed by the probabilistic model rather than learned within a generic hypothesis class. Our approach achieves a better performance compared to shallow OT, deep domain adaptation and neural OT baselines on benchmarks spanning audio, signal processing and visual adaptation modalities.