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

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.

• The paper shows that naive adaptive routing collapses to ensemble averaging, hindering effective cross-domain EEG decoding.
• It introduces structural fixes—symmetric anchor, frozen query encoder, and decoupled key alignment—to enable genuine, sample-specific routing.
• Empirical tests on three EEG datasets demonstrate significant proxy gaps and robust domain specialization without the need for calibration.

Dynamic Subspace Routing on the Stiefel Manifold for Cross-Domain EEG Decoding

Problem Formulation and Motivation

The challenge of cross-domain EEG decoding arises from significant inter-subject variability in EEG signals: subject-specific factors drive covariance matrices from different individuals into consistently distinct locations on the symmetric positive definite (SPD) manifold. Traditional Riemannian deep learning methods, especially those using SPDNet architectures, exploit SPD geometry within domains but apply a shared subspace projection (BiMap) across all subjects. This mismatch severely limits cross-domain generalization since a single learned subspace cannot account for heterogeneity across the SPD manifold's regions associated with distinct domains. Existing adaptation solutions often require target domain calibration or fail to yield generalizable, sample-specific subspace selection. The central question addressed is: under what conditions can adaptive, sample-conditioned subspace selection—implemented as a routing mechanism on the Stiefel manifold—achieve genuine, domain-structured specialization without collapsing to a degenerate ensemble or requiring calibration?

Fundamental Insights: Degeneracy of Naive Adaptive Routing

The proposed approach organizes $K$ expert projection filters on the Stiefel manifold, with each expert hypothesized to specialize to a particular region/domain of the SPD space. For each input covariance, a routing mechanism assigns attention weights (softmax over dot products between a learned query and expert keys), interpolating among expert filters via weighted Riemannian barycenters. However, the authors prove that naive adaptive routing is fundamentally degenerate: absent architectural constraints or inductive biases, routing weights collapse to near-uniform distributions, and the layer reduces to ensemble averaging—identical for all samples and equivalent to a fixed filter in effect. This collapse arises from a self-reinforcing dynamic: when experts are similar, routing signals (gradients) lack distinguishing power, preventing differentiation among experts. Experimentally, the gain over a single baseline filter results only from ensemble diversity, not any genuine sample-specific adaptation, as evidenced by a null K=1 proxy gap (the accuracy difference between the adaptive model and its K=1 ensemble proxy).

Structural Solutions: Conditions for Genuine Committed Routing

Building from a theoretical analysis and motivated by parallels to continual-learning prompting strategies (notably L2P), the authors show that three architectural modifications are necessary and sufficient to break the degeneracy:

1. Symmetric Anchor ($W_\text{base}$): Decoupling the reference point for tangent-space interpolation from any expert removes proximity bias and enables symmetric competition among experts.
2. Frozen Domain-Discriminative Query Encoder (DSP): The query mechanism for routing is restructured to use features extracted via a fixed, between-domain separation projection of the log-covariance tangent space; this ensures routing reflects domain (not task) structure and is analogized to a frozen backbone in prompting.
3. Decoupled Key Alignment Loss: Keys are detached from the task loss and trained purely by an alignment objective driving each expert's key toward stable query attractors, using a winner-take-all regime to prevent key collapse.

A diagnostic framework—comprising the K=1 proxy gap, routing entropy (commitment degree), and alignment ratio (domain-structured routing)—is employed to decisively measure whether the routing mechanism is genuinely specialized.

Empirical Evaluation and Results

Experiments span three motor imagery EEG datasets—Weibo2014 (high-dimensional, low-domain-count), BNCI2015001 (small dimension, many domains), BNCI2014001 (intermediate settings)—that rigorously test both high and low tangent-to-domain (ρ) regimes. The structural fixes together (the Domain-Adaptive Stiefel Pool (DASP) layer) yield consistent and substantial positive K=1 proxy gaps: on Weibo2014, balanced accuracy increases from 0.773 (baseline) to 0.823 (DASP), with genuine routing validated by a proxy gap of +0.029. For BNCI2015001 and BNCI2014001, increases of +0.247 and +0.170 are observed, with nearly all folds in cross-validation showing statistically significant positive proxy gaps across the diagnostic framework. In all settings, committed, domain-structured routing is demonstrated by lower entropy and higher alignment ratios.

Ablations confirm that without all three architectural components, routing remains degenerate. Notably, introducing only a symmetric anchor or DSP is insufficient; key decoupling further sharpens domain structure and model commitment.

A key methodological contribution is a unified scaling rule for employing DSP and alignment losses, based solely on the tangent-to-domain ratio ρ, eliminating dataset-specific hyperparameter search. The number of experts $K$ is shown to scale sub-linearly with the number of domains $D$.

Theoretical and Architectural Extensions

The study outlines a full-rank rectangular extension: moving beyond Stiefel (orthonormal) constraints, experts can be parameterized via a pseudo-polar decomposition or unconstrained matrices with log-determinant barriers, supporting amplitude scaling as well as rotation and enabling modeling of non-preconditioned (heteroscedastic) domain differences.

Future directions explored involve top-p sparse routing (hard attention across experts), transductive adaptation to unseen domains (e.g. in LOSO protocols), Riemannian-geometric nearest-neighbour selection for initializing embeddings in zero-shot settings, and per-domain CSP targets for enhanced alignment supervision.

Implications and Future Outlook

This work establishes that without correct architectural and optimization principles, adaptive expert routing on the Stiefel manifold is ineffective for cross-domain EEG decoding, and may mislead performance interpretation due to ensemble effects. The DASP layer and associated diagnostics present a robust, practical tool for domain-aware subspace adaptation, scalable across divergent dataset geometries. The theoretical insight that only architecture- and loss-level decoupling can elicit specialization has broader significance for mixture-of-experts, adaptive routing, and modular neural architectures on non-Euclidean spaces.

Practically, this approach enables calibration-free, subject-adaptive BCI pipelines, with concrete gains in cross-domain decoding accuracy, and does so with zero dataset-specific tuning. Theoretically, it opens lines of inquiry for general adaptive routing over Riemannian manifolds—potentially applicable to cross-modality domains or heterogeneous sensor fusion settings.

Open problems include architecture choices for fully general expert routing (rectangular/factorized), richer sample-efficient adaptation in the absence of explicit domain labels, and tighter geometric alignment between routing spaces and semantic domain structure.

Conclusion

This study rigorously characterizes the failure of naive adaptive subspace routing on the Stiefel manifold for EEG domain adaptation, providing precise diagnostics and actionable architectural guidance to realize genuinely committed and domain-structured routing. The DASP layer delivers substantive accuracy gains without calibration data or hyperparameter search, establishing a new baseline and methodology for future cross-domain decoding and adaptive mixture-of-expert research in Riemannian deep learning.