  The machine learning community has shown increasing interest in addressing
the domain adaptation problem on symmetric positive definite (SPD) manifolds.
This interest is primarily driven by the complexities of neuroimaging data
generated from brain signals, which often exhibit shifts in data distribution
across recording sessions. These neuroimaging data, represented by signal
covariance matrices, possess the mathematical properties of symmetry and
positive definiteness. However, applying conventional domain adaptation methods
is challenging because these mathematical properties can be disrupted when
operating on covariance matrices. In this study, we introduce a novel geometric
deep learning-based approach utilizing optimal transport on SPD manifolds to
manage discrepancies in both marginal and conditional distributions between the
source and target domains. We evaluate the effectiveness of this approach in
three cross-session brain-computer interface scenarios and provide visualized
results for further insights. The GitHub repository of this study can be
accessed at
https://github.com/GeometricBCI/Deep-Optimal-Transport-for-Domain-Adaptation-on-SPD-Manifolds.
