Routing on the Stiefel Manifold: When Does Adaptive Subspace Selection Help for Cross-Domain EEG Decoding?
This work is significant for researchers and practitioners in EEG signal processing, offering a method to improve cross-domain EEG decoding without requiring target-domain calibration data or learning subject-specific components that do not generalize.
This paper addresses the challenge of cross-domain EEG decoding where covariance matrices from different subjects occupy distinct regions of the SPD manifold. The authors propose dynamic Stiefel routing, a system of K expert projection filters on the Stiefel manifold, each specialized for a different region of the SPD manifold, with inputs routed to the most appropriate filter. This approach yields consistent gains across three datasets, improving balanced accuracy from 0.773 to 0.823, 0.757 to 0.809, and 0.801 to 0.839.
Cross-domain EEG decoding remains challenging despite advances in Riemannian deep learning: covariance matrices from different subjects occupy systematically distinct regions of the SPD manifold, yet existing domain adaptation methods either require target-domain calibration data or learn subject-specific components that cannot generalise across domains. We propose dynamic Stiefel routing: a pool of $K$ expert projection filters on the Stiefel manifold, each specialised for a different region of the SPD manifold, with each input covariance routed to the most appropriate filter via cross-attention, adapting the subspace projection per sample. A central finding is that this approach, implemented naively, provably collapses to ensemble averaging: when routing weights are uniform, the adaptive filter reduces exactly to an equal-contribution combination of experts, indistinguishable from a single fixed filter. Three structural properties break this degeneracy: a symmetric anchor $W_{\mathrm{base}} \in \mathrm{St}(n,k)$ that removes proximity bias among experts; a frozen domain-discriminative query encoder that decouples routing from task optimisation; and a decoupled key alignment loss that trains expert keys toward stable domain attractors. Together they produce the first genuinely committed and domain-structured routing on SPD manifolds, with consistent gains across three datasets: balanced accuracy improves from $0.773\to 0.823$, $0.757\to 0.809$, and $0.801\to 0.839$, with the alignment strategy determined automatically by a single data-driven rule and no dataset-specific hyperparameter search.