Deep Optimal Transport for Domain Adaptation on SPD Manifolds

In recent years, there has been significant interest in solving the domain adaptation (DA) problem on symmetric positive definite (SPD) manifolds within the machine learning community. This interest stems from the fact that complex neurophysiological data generated by medical equipment, such as electroencephalograms, magnetoencephalograms, and diffusion tensor imaging, often exhibit a shift in data distribution across different domains. These data representations, represented by signal covariance matrices, possess properties of symmetry and positive definiteness. However, directly applying previous experiences and solutions to the DA problem poses challenges due to the manipulation complexities of covariance matrices.To address this, our research introduces a category of deep learning-based transfer learning approaches called deep optimal transport. This category utilizes optimal transport theory and leverages the Log-Euclidean geometry for SPD manifolds. Additionally, we present a comprehensive categorization of existing geometric methods to tackle these problems effectively. This categorization provides practical solutions for specific DA problems, including handling discrepancies in marginal and conditional distributions between the source and target domains on the SPD manifold. To evaluate the effectiveness, we conduct experiments on three publicly available highly non-stationary cross-session brain-computer interface scenarios. Moreover, we provide visualization results on the SPD cone to offer further insights into the framework.