Organizations: BIFOLD, Germany · Technical University of Berlin, Germany · Advanced Telecommunications Research Institute International, Japan · RIKEN AIP, Japan · Nanyang Technological University, Singapore
Electroencephalography (EEG) based brain-computer interfaces enable direct brain-to-device communication for applications such as rehabilitation and communication. However, their practical utility is often limited as the non-stationary nature of the EEG data introduces distribution shifts across domains (e.g., sessions and subjects). Adapting machine learning models to be invariant to these shifts in an unsupervised way, without using costly labeled calibration data, would drastically improve the utility of EEG data. In this work, we use a classic generative model of EEG to study distribution shifts introduced by the domain-specific forward process, which is associated with factors such as head geometry. We theoretically show that such distribution shifts can be recovered solely through linear transformations on the Symmetric Positive Definite manifold. Building on this insight, we propose SPDAlign, an interpretable framework for promoting domain-invariant EEG learning. SPDAlign first aligns the domain-specific means and corrects global rotations across domains using a recent optimal transport technique called Wasserstein Procrustes. We systematically study the proposed approach through simulations and demonstrate its competitive performance on extensive public EEG datasets. Additionally, SPDAlign is a globally linear framework and is intrinsically interpretable, so that the framework can identify frequency ranges of interest, determine the spatial patterns reflecting source-sensor relationships, and address cross-subject variability.
Figures & tables
Figure 1: SPDAlign Overview. (a) Architecture: an end-to-end Riemannian learning framework on the SPD manifold that aligns domain-specific means and applies orthogonal transformations across domains for multi-source, multi-target unsupervised domain adaptation. (b) Rotational alignment: Global rotation shifts between domains are resolved by estimating an orthogonal transformation Q via Wasserstein Procrustes. (c) t-SNE visualization: Classification space for the driving fatigue dataset across all subjects. Color represents labels, and shape denotes domains.
Figure 2: SPDAlign architecture. SPDAlign optimizes parameters Θ={θ,ϕ,ψ} in an end-to-end fashion: (i) a feature extractor fθ maps raw EEG signals onto the SPD manifold SD+ ; (ii) the SPDAlign module mϕ addresses shifts by aligning means and correcting rotations ( Eqs. 11 and 12 ); and (iii) a shared linear classifier gψ outputs predictions y^ . By maintaining global linearity, SPDAlign is equivalent to the Tangent Space Mapping framework, providing interpretability.
Figure 3: SPDAlign steps toward domain-invariant representations. The t-SNE visualization illustrates the alignment process for the workload dataset in the classification space. (a) Moment alignment: aligning the domain-specific Fréchet means ( Eq. 11 ) to the identity matrix (previous work). (b) SPDAlign and (c) SPDAlign (Oracle): applying the orthogonal transformation Q estimated via Wasserstein Procrustes ( Eq. 9 ) and empirical risk minimization. SPDAlign enhances class separability, addressing forward modeling shifts without requiring labels. The remaining gap in the unsupervised setting is primarily due to suboptimal rotation estimation.
Figure 4: SPDAlign interpretability. Patterns extracted from the workload dataset and class low (cross-subject) are shown above. The top-left panel details the relative contribution of each extracted latent source. For clarity, only the two most discriminative sources are displayed in the left column panels. The top row panels summarize the frequency profile of each spectral channel (derived from the output of the four temporal convolution layers in fθ ). Topographic plots show the projection of source activity to regions covered by the EEG channels; areas shaded in darker blue or red indicate greater discriminative source activity. These results align with prior research, as the frontal and parietal areas and the theta band serve as biomarkers of mental workload ( Borghini et al., 2014 ) .
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Simulation performance evaluation across varying class separability. The panels illustrate the balanced accuracy ( y -axis) of different alignment methods as a function of the rotational shift in degrees ( x -axis) introduced by domain-specific forward modeling shifts. Each panel represents a distinct level of class separability, with the degree of class overlap visualized in the gray insets. We compare the performance of SPDAlign against recentering to the identity (RCT) ( Zanini et al., 2017 ) . The blue line indicates source domain performance, providing an empirical upper bound for the UDA task.
Figure 6: Simulation performance over the number of data samples. Balanced accuracy scores (higher is better) across rotation degree on the x-axis and number of data samples across panels. Larger sample sizes lead to a clearer advantage of SPDAlign over RCT under moderate rotational shifts.
Figure 7: Simulation performance over the number of informative sources. Balanced accuracy scores (higher is better) across rotation degree on the x-axis and number of informative sources across panels. SPDAlign generally remains more robust than RCT; the advantage is particularly evident when sufficient informative source structure is available, supporting the robustness of SPDAlign under typical EEG regimes.
Figure 8: Simulation performance over label shift ratios. Balanced accuracy scores (higher is better) across rotation degree on the x-axis. As expected, label shifts degraded performance, but UOT, which relaxes the marginal constraints, still provided performance gains, indicating that UOT is a suitable choice when class imbalance is present.
Figure 9: Simulation performance over available data percentage. Balanced accuracy scores (higher is better) across rotation degree on the x-axis. With limited incoming data, the estimation is less reliable, resulting in a smaller advantage over RCT. As more target samples become available, SPDAlign consistently shows advantages over RCT.
Figure 10: Simulation performance over the number of data dimensions. Balanced accuracy scores (higher is better) across the number of data dimensions on the x-axis. As the proportion of noise sources increases (i.e., as SNR decreases), all methods’ performance declines. SPDAlign shows a clear advantage under mild noise conditions, remains beneficial under moderate noise, and maintains similar performance without degrading results under more extreme noise levels.
Figure 11: Simulation performance over the number of source domains. Balanced accuracy scores (higher is better) across the number of source domains on the x-axis. In the absence of source domain rotation correction, training performance degrades as the number of source domains increases, due to the accumulation of conflicting domain-specific rotation shifts.
πt=argminπ∈Π(a,b)∑i,jπij∥Qtxi−yj∥22
Appendix
Algorithm 2 Wasserstein Procrustes in SPDAlign
Figure 12: SPDAlign interpretation results. Patterns extracted from the workload dataset and class high (cross-subject setting).
Electroencephalography (EEG) based Brain-Computer Interfaces (BCIs) often require unsupervised domain adaptation (UDA) to generalize across subjects and sessions. While Riemannian alignment methods like the Riemannian Centering Transformation (RCT) are effective for handling covariate shifts, they implicitly assume balanced class priors. However, in realistic online BCI scenarios, the label distributions vary dynamically (label shift), causing standard alignment techniques to geometrically misalign the target data distributions. In this work, we propose OSPDIM (Online SPD manifold information maximization), a source-free online UDA framework designed to address label shifts on the Riemannian manifold. OSPDIM introduces a manifold-constrained bias parameter into the tangent space mapping, which is optimized via information maximization to correct the geometric skew caused by imbalanced data streams. Unlike offline methods relying on global batch statistics, OSPDIM estimates and corrects geometric bias on-the-fly. Simulations on 2D SPD matrices visually demonstrate that OSPDIM successfully rectifies the misalignment where standard centering fails. Extensive experiments on multiple motor imagery datasets show that OSPDIM significantly outperforms standard Riemannian baselines, particularly in challenging online adaptation scenarios with severe class imbalance, offering a robust solution for practical, plug-and-play BCI systems.
Shiwen Chu, Shanglin Li, Motoaki Kawanabe +1
Advanced Telecommunications Research Institute International, 2-2-2 Hikaridai Seika-cho, Soraku-gun, Kyoto, Japan · BIFOLD – Berlin Institute for the Foundations of Learning and Data, 10587 Berlin, Germany · Nara Institute of Science and Technology, 8916-5 Takayama-cho, Ikoma, Nara, Japan
Cross-subject EEG decoding promises more training data, but it also exposes neural networks to strong inter-subject distribution shifts. We study whether task supervision and architecture alone can learn subject-aligned representations. We replace a shared EEG encoder with subject-specific encoders followed by a common classifier, and compare this hybrid model with standard EEGNet, AttentionBaseNet, and CTNet baselines with Euclidean Alignment (EA) on three motor-imagery datasets and one motor-execution dataset. EA improves shared encoders by recentering subject covariances, whereas the hybrid encoder reduces reliance on EA: removing EA has little effect on validation-loss dynamics or latent-space organization, and both hybrid variants consistently outperform non-aligned shared baselines. Subject-specific heads increase class distinctiveness and place each subject close to its own latent manifold while improving within-subject class separation. However, on cross-subject classification, subject-specific heads hinder direct parameter transfer to unseen subjects, motivating quantitative head selection and a brief calibration session. Although decoding gains depend on the dataset and backbone, our main findings concern that the sole use of architecture pressure promotes representation learning and alignment in a direction complementary to domain adaptation methods such as Euclidean Alignment. A per-subject low-rank adapter of only 2Cr parameters recover the full encoder's accuracy across five backbones and ranks r=1 to 16, so the per-subject module can be compressed by two to three orders of magnitude.
Bruna J. Lopes, Gabriel Schwartz, Sylvain Chevallier +2
University of São Paulo, São Paulo, Brazil · Université Paris-Saclay, Inria TAU team, LISN-CNRS, France · Institut de neuromodulation, GHU Paris, psychiatrie et neurosciences, centre hospitalier Sainte-Anne, pôle hospitalo-universitaire 15, Université Paris Cité, Paris, France +3
Cross-subject motor imagery decoding remains a fundamental challenge in EEG-based brain-computer interfaces due to substantial inter-subject variability. Recent approaches have leveraged Riemannian geometry by representing EEG signals as covariance matrices on the symmetric positive definite (SPD) manifold. However, existing methods primarily focus on manifold-based representations while largely overlooking subject-specific variations in covariance dispersion and orientation. In this work, we address these challenges through geometry-aware congruence transformations and propose three complementary models: (i) Discriminative Congruence Transform (DCT), (ii) Deep Linear DCT (DLDCT), and (iii) Deep DCT-UNet (DDCT-UNet). The proposed models are evaluated both as manifold alignment modules for downstream classifiers and as end-to-end discriminative architectures optimized via cross-entropy with a custom logistic regression head. Experiments on challenging cross-subject motor imagery benchmarks demonstrate consistent improvements in transductive decoding performance, achieving 2-3% higher accuracy than strong baselines. These results highlight the effectiveness of geometry-aware congruence learning for mitigating inter-subject variability in EEG decoding.
Sanjeev Manivannan, Chandra Shekar Lakshminarayan
Department of Biotechnology · Department of Data Science and Artificial Intelligence