Geometry-Aware Convolutional Neural Networks on SPD Manifolds

Deep learning architectures typically assume Euclidean geometry in their inputs, an assumption that proves inadequate for electroencephalography (EEG) signals represented as covariance matrices on the manifold of symmetric positive definite (SPD) matrices. This work introduces a geometry-aware framework RCEEGnet that integrates Riemannian principles with convolutional neural networks (CNNs) to respect the manifold nature of EEG covariance data. By projecting covariance matrices onto the Stiefel manifold and mapping them to a tangent space for classification, the proposed model preserves the intrinsic geometric constraints of SPD data. This approach fosters interpretability by capturing inter-channel dependencies and demonstrates how geometry-aware deep learning can advance EEG-based brain-computer interface (BCI) applications. Potential extensions include leveraging alternative Riemannian metrics and expanding manifold-aware layers to enhance broader machine learning frameworks.