Deep Optimal Transport for Domain Adaptation on SPD Manifolds

The domain adaptation (DA) problem on symmetric positive definite (SPD) manifolds has raised interest in the machine learning community because of the growing potential for the SPD-matrix representations across many cross-domain applicable scenarios. However, due to the different underlying space, the previous experience and solution to the DA problem cannot benefit this new scenario directly. This study addresses a specific DA problem: the marginal and conditional distributions differ in the source and target domains on SPD manifolds. We then formalize this problem from an optimal transport perspective and derive an optimal transport framework on SPD manifolds for supervised learning. In addition, we propose a computational scheme under the optimal transport framework, Deep Optimal Transport (DOT), for general computation, using the generalized joint distribution adaptation approach and the existing Riemannian-based network architectures on SPD manifolds. DOT is applied to the real-world scenario and becomes a specific EEG-BCI classifier against the cross-session motor-imagery classification from the calibration phase to the feedback phase. In the experiments, DOT exhibits a marked improvement in the average accuracy in two highly non-stationary cross-session scenarios in the EEG-BCI classification, respectively, indicating the proposed methodology's validity.