Stiefel-SPD Manifold Graph Convolution for End-to-End EEG Learning

Electroencephalographic (EEG) decoding relies heavily on second-order (covariance) structure that lives on the manifold of symmetric positive-definite (SPD) matrices. Conventional deep networks in Euclidean space ignore this geometry, distorting geodesic relations between covariances; classical Riemannian pipelines respect SPD metrics but typically use fixed projections and a single global tangent embedding, which limits task adaptivity and incurs cubic costs in the channel dimension. We propose a fully geometry-consistent architecture that preserves manifold structure end-to-end while remaining trainable at scale. A compact depthwise-separable convolutional neural network (CNN) produces features whose regularized covariances lie on the SPD manifold. A learnable orthonormal projection, optimized on the Stiefel manifold via Riemannian stochastic gradient descent (SGD) with QR-factorization (QR) retraction, reduces dimensionality without breaking positive-definiteness and preserves an eigenvalue floor. We then perform tangent space graph-SPD aggregation on a scalp <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-nearest-neighbor graph—neighbor covariances are transported to the reference tangent space, attention-averaged, and mapped back via the exponential—followed by a log-Euclidean mapping and linear softmax classification. This Stiefel<inline-formula> <tex-math notation="LaTeX">$\!\to $ </tex-math></inline-formula>Graph-SPD<inline-formula> <tex-math notation="LaTeX">$\!\to \log $ </tex-math></inline-formula> chain explains why full geometric consistency matters: it avoids Euclidean shortcuts, keeps all intermediates SPD, and makes log/exp costs cubic in the reduced rank <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>. In cross-subject evaluation on three public datasets, the model attains <inline-formula> <tex-math notation="LaTeX">${83}.{2}\%\!/\!{81}.{5}\%\!/\!{79}.{7}\%$ </tex-math></inline-formula> accuracy with improved macro-<inline-formula> <tex-math notation="LaTeX">${F}_{{1}}$ </tex-math></inline-formula>, strong separability (macro-AUROC <inline-formula> <tex-math notation="LaTeX">$\approx {0}.{90}$ </tex-math></inline-formula>), and well-calibrated probabilities (ECE <inline-formula> <tex-math notation="LaTeX">$\le {0}.{04}$ </tex-math></inline-formula>), outperforming strong Euclidean CNNs and Riemannian baselines while remaining computationally pragmatic.