Routing on the Stiefel Manifold: When Does Adaptive Subspace Selection Help for Cross-Domain EEG Decoding?

2026-05-29

AI Summary

This work addresses the challenge of large inter-subject distribution shifts in covariance matrices on the SPD manifold for cross-subject EEG decoding, where existing domain adaptation methods either rely on target-domain calibration data or exhibit limited generalization. The authors propose a dynamic Stiefel routing mechanism that constructs multiple expert projection filters on the Stiefel manifold and employs cross-attention to enable sample-level adaptive subspace selection. Innovatively integrating symmetric anchors, a frozen domain-discriminative query encoder, and a decoupled key alignment loss, this approach realizes—for the first time—a domain-structure-aware routing mechanism on the SPD manifold, effectively avoiding degeneration into simple ensemble averaging. Evaluated on three benchmark datasets, the method achieves balanced accuracies of 0.823, 0.809, and 0.839, respectively, without requiring dataset-specific hyperparameter tuning.

Abstract

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.

Problem

• cross-domain EEG decoding
• SPD manifold
• domain adaptation
• covariance matrices
• generalisation

Innovation

• Stiefel manifold
• dynamic routing
• cross-domain EEG decoding
• SPD manifold
• domain adaptation