PaperlayerView exact revision on arXiv

Source-linked AI summary

SPDAlign: Interpretable Riemannian Alignment for EEG Forward Modeling Shifts

Shanglin Li, Shiwen Chu, Okan Koç, Chenyu Liu, Qibin Zhao, Motoaki Kawanabe, Mitsuo Kawato, Yi Ding

arXiv:2610.06315v1cs.LG

TL;DR

EEG-based brain–computer interfaces are limited by distribution shifts across subjects, while labeled calibration challenges scalability. SPDAlign uses theoretically motivated SPD-manifold alignment to address forward-model shifts, and achieves state-of-the-art results on three of four benchmark datasets.

  • Problem

    EEG-based brain–computer interfaces generalize poorly across subjects, and labeled calibration with domain-specific models poses scalability challenges.

    Source: §1 · p. 1

  • Method

    SPDAlign theoretically motivates compensating for forward-model shifts through domain-mean alignment and global rotations on the SPD manifold, without target labels during inference.

    Source: §1 · p. 2, §3.2 · p. 5, §3.1 · p. 5

  • Results

    Three of four benchmark datasets: SPDAlign achieved state-of-the-art results, and its workload case study identified discriminative source activity peaking in frontal and parietal theta.

    Source: §1 · p. 2, §5 · p. 9

  • Takeaways & Limitations

    SPDAlign's globally linear framework can identify frequency ranges and spatial source–sensor patterns while addressing cross-subject variability.

    Source: §1 · pp. 2–3

    Without label supervision, Wasserstein Procrustes can infer semantically mismatched correspondences when class-conditional structures overlap or are weakly separated, biasing rotation estimates.

    Source: §4.1 · p. 8

Abstract

from arXiv · show

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.

1 INTRODUCTION

EEG-based brain–computer interfaces face poor cross-subject generalization, while labeled calibration challenges scalability. SPDAlign grounds unsupervised domain alignment in EEG forward modeling and uses globally linear transformations to support domain-invariant learning and interpretation.

1 INTRODUCTION

  • EEG-based BCIs generalize poorly across subjects, and labeled calibration with domain-specific models poses scalability challenges; unsupervised domain adaptation seeks transfer without target labels.Cross-subject transfer is described as a grand challenge in EEG research.Source: §1 · p. 1
  • Existing heuristic alignment methods address statistical and rotational discrepancies but lack theoretical foundations, limiting their explanatory power.The cited methods align first- and second-order statistics as well as rotational components.Source: §1 · p. 2
  • EEG forward-model shifts can be compensated on the SPD manifold through domain-mean matching followed by a global rotation, recovering robust latent source-covariance representations.The shifts arise from domain-specific forward matrices associated with variations such as head geometry.Source: §1 · p. 2
  • SPDAlign aligns domain-specific means, then estimates global rotations with source labels or Wasserstein Procrustes for unlabeled targets in multi-source, multi-target adaptation.Unlike the cited prior frameworks, it does not require target-domain labels.Source: §1 · p. 2
  • Simulations and public EEG datasets show competitive cross-subject decoding; the workload case study identifies discriminative frontal and parietal theta activity consistent with reported workload markers.The evaluations span motor imagery, workload, mental stress, and driving fatigue.Source: §1 · p. 2

2 PRELIMINARIES

The preliminaries define unsupervised adaptation from labeled source domains to unlabeled targets, then motivate covariance-based modeling of EEG and geometric alignment on the SPD manifold. Wasserstein Procrustes provides a way to jointly estimate rotations and unknown correspondences.

2.1 MULTI-SOURCE MULTI-TARGET UDA

  • Multi-source, multi-target UDA learns from labeled source datasets to generalize to unseen, unlabeled target domains that share the same input and label spaces.Source: §2.1 · p. 3

2.2 EEG FORWARD MODELING

  • EEG epochs are modeled as linear mixtures of informative latent sources and additive noise, with an invertible domain-specific forward model.The task label depends on latent-source spatial covariance, and observed sensor covariances are sufficient statistics for decoding when the forward matrix is invertible.Source: §2.2 · p. 3

2.3 SPD LEARNING IN EEG

  • SPD learning represents EEG epochs with covariance matrices, estimates their Fréchet mean, maps them to its tangent space, and recenters them at the identity for classification.Source: §2.3 · pp. 3–4

2.4 WASSERSTEIN PROCRUSTES

  • Wasserstein Procrustes jointly estimates an orthogonal alignment and unknown point correspondences by combining Wasserstein distance with orthogonal Procrustes.Orthogonality preserves pairwise distances, restricting the transformation to a global rotation; exact solutions are available for each sub-optimization when the other variable is fixed.Source: §2.4 · p. 4

3 METHODS

The methods model EEG domain shifts through domain-specific forward matrices and show how mean alignment followed by a global rotation can recover robust latent representations. SPDAlign implements these steps for source training and unlabeled target inference while retaining a globally linear, interpretable model.

3.1 EEG FORWARD MODELING SHIFTS

  • The shift model assumes observed-data marginals vary across domains while the conditional source-to-label relationship remains invariant, with forward matrices capturing shifts such as those associated with head geometry.Polar decomposition expresses each forward matrix as a domain-specific transition and an orthogonal matrix, linking the shifts to translations and rotations on the SPD manifold.Source: §3.1 · pp. 4–5
  • Aligning domain-specific Fréchet means to identity and then applying a global rotation can recover robust latent representations that retain a linear relationship to labels.The rotations use labeled-source correspondences during training and Wasserstein Procrustes without labels during target inference.Source: §3.1 · p. 5, §3.2 · p. 5

3.2 SPDALIGN FRAMEWORK

  • SPDAlign trains a shared feature extractor, alignment module, and linear classifier end to end, using source labels for Procrustes alignment and unlabeled target data for Wasserstein Procrustes.Training first learns features without rotation alignment, then freezes the feature extractor, estimates rotations, and fine-tunes the classifier.Source: §3.2 · pp. 5–6, Figure 2 · p. 6
  • SPDAlign retains global linearity, allowing classifier weights and learned representations to be mapped to sensor-space spatial patterns that reflect source–sensor relationships and source contributions.The paper identifies this interpretability as useful for determining brain regions and frequencies associated with cognitive processes.Source: §3.2 · p. 6

4 EXPERIMENTS

Across four EEG tasks, SPDAlign achieves state-of-the-art performance on three and provides interpretable workload patterns. Simulations, validation, and ablations support rotation alignment while identifying limits of unsupervised rotation estimation.

4 EXPERIMENTS

  • In simulations, SPDAlign outperforms RCT across scenarios, with robustness under typical EEG conditions of moderate sample sizes, low SNR, and informative sources.Performance declines as rotational shifts increase; the simulations vary class separability, sample size, dimensionality, source count, informative sources, data availability, and label shifts.Source: §4 · p. 6

4.1 MAIN EXPERIMENTS

  • SPDAlign achieves state-of-the-art performance on three of four EEG tasks in cross-subject decoding.Table 1 reports grand-average test-set balanced accuracy and standard deviation; its caption identifies SPDAlign (Oracle) as an empirical upper bound using target labels only to estimate the optimal rotation.Source: Table 1 · p. 7, §4.1 · p. 7
  • The evaluation compares cross-subject decoding across four public datasets with deep-learning, foundation, geometric, and UDA baselines, plus a label-assisted Oracle variant.The Oracle uses target labels only for optimal rotation estimation; the rest follows the standard source-ERM protocol.Source: §4.1 · p. 7, Table 1 · p. 7
  • Workload interpretations identify a top contributing source peaking in the theta band and discriminative activity primarily in frontal and parietal areas.Figure 4 displays only the two most discriminative sources; its topographic plots map source activity to EEG-channel regions.Source: §4.1 · pp. 7–8, Figure 4 · p. 9
  • The Oracle results support forward modeling shifts as a primary source of EEG distribution shifts, with the remaining unsupervised-to-oracle gap attributed mainly to rotation-estimation error.The Oracle uses target labels only to estimate the optimal rotation; the classifier remains fixed and shared across domains.Source: §4.1 · p. 8, Figure 3 · p. 8
  • Wasserstein Procrustes can yield biased rotations when overlapping or weakly separated class structures produce ambiguous correspondences; performance also degrades under extreme rotations or limited class separability.Erroneous correspondences may accumulate during iterative optimization, leaving a gap relative to the ERM version.Source: §4.1 · p. 8
  • The full configuration performs best; removing rotation alignment significantly reduces performance (p = 0.02), while omitting both alignment components causes the largest accuracy decrease.The implementation uses a global reference mean across domains; partial moment alignment is otherwise implicitly enforced, reducing the apparent impact of explicit moment alignment.Source: §4.1 · p. 8, Table 2 · p. 9

5 DISCUSSIONS

The discussion presents SPDAlign as a globally linear Riemannian framework for multi-source/multi-target UDA, with competitive performance and interpretable outputs. It notes computational limits for high-dimensional SPD features and possible extensions to online UDA.

5 DISCUSSIONS

  • SPDAlign combines global linearity and Riemannian learning for multi-source/multi-target UDA, mitigating forward-modeling shifts through unsupervised SPD-manifold alignment.The discussion reports state-of-the-art results on three of four benchmark datasets and interpretable identification of frequency ranges and brain regions.Source: §5 · p. 9
  • Eigendecompositions impose computational costs that limit SPDAlign’s application to high-dimensional SPD features.Source: §5 · p. 9
  • The study evaluates offline UDA; the authors suggest online extension via running-statistic tracking and online Wasserstein-distance estimation.They also identify more robust rotation estimation and online BCI applications as future directions.Source: §5 · pp. 9–10

A SPD MANIFOLD IN A NUTSHELL

The appendix introduces the SPD manifold and its affine-invariant geometry, including tangent-space maps and Fréchet means. Its related-work discussion distinguishes SPDAlign by adding unsupervised rotation estimation to multi-source, multi-target EEG adaptation.

A SPD MANIFOLD IN A NUTSHELL

  • The SPD manifold comprises symmetric positive definite matrices and is treated as a Riemannian manifold with Euclidean tangent spaces that locally approximate its distances.Source: A SPD MANIFOLD IN A NUTSHELL · p. 16
  • The study adopts the affine-invariant Riemannian metric, with distance between SPD matrices expressed using a matrix logarithm and the Frobenius norm.The passage identifies the matrices, matrix logarithm, trace, and Frobenius norm used in the distance expression.Source: A SPD MANIFOLD IN A NUTSHELL · p. 16
  • A Fréchet mean minimizes the average squared distances to a set of SPD matrices; weighted means lie along the geodesic connecting two points.The passage gives γ = 0.5 as the selected weight and describes γ in [0, 1] for weighted means.Source: A SPD MANIFOLD IN A NUTSHELL · p. 16
  • The logarithmic map projects manifold points to a tangent space, while the exponential map performs the inverse projection.Source: A SPD MANIFOLD IN A NUTSHELL · p. 16

A.1 RELATED WORK

  • Prior end-to-end alignment tracked domain-specific running Fréchet means, whereas SPDAlign also accounts for rotation transformations.The related-work discussion says prior rotation methods relied on target labels, while SPDAlign estimates EEG rotations without label access and addresses multi-source, multi-target adaptation.Source: §A.1 · p. 16

B DERIVATION OF DOMAIN-INVARIANT LATENT REPRESENTATIONS

The derivation uses Fréchet-mean properties to simplify covariance relationships, then combines recentering and global rotation transformations to obtain a domain-invariant representation.

B DERIVATION OF DOMAIN-INVARIANT LATENT REPRESENTATIONS

  • Congruence invariance of the Fréchet mean and the linear relation between latent and empirical covariances support the derivation's covariance simplification.The passage introduces this step before the subsequent identity substitution.Source: B DERIVATION OF DOMAIN-INVARIANT… · p. 17
  • The derivation invokes a proposition that each latent covariance Fréchet mean converges to the identity as L → ∞.The resulting identity is substituted into the preceding expression to simplify it.Source: B DERIVATION OF DOMAIN-INVARIANT… · p. 17
  • Combining recentering, domain-specific global rotations, and the cited covariance relations yields the derivation's final transformed representation.The passage connects the two transformations to the preceding equations.Source: B DERIVATION OF DOMAIN-INVARIANT… · p. 17

C SIMULATIONS

Simulations test SPDAlign against recentering to the identity (RCT) under varied rotational shifts and forward-model conditions. SPDAlign is generally more robust, with its advantage depending on data and noise conditions.

Simulation setup

  • The simulations generate log-space features with one label-relevant dimension, map them to the SPD manifold, and project them through domain-specific forward matrices.Data are divided among source and target domains, with 500 observations per domain; rotation shifts are introduced with angle-parameterized or random orthogonal matrices.Source: C SIMULATIONS · p. 18

Class separability

  • SPDAlign remains more accurate than RCT across a wider range of rotations, while greater class separability improves robustness for both methods.Moment alignment alone compensates for forward-model transition effects when there is no rotation; source-domain performance is the empirical UDA upper bound.Source: C SIMULATIONS · p. 18, Figure 5 · p. 19

Sample size

  • With moderate sample sizes, SPDAlign retains a clear advantage over RCT even under severe rotations; fewer samples weaken covariance and Wasserstein Procrustes estimation.Larger sample sizes make the advantage clearer under moderate rotational shifts.Source: C SIMULATIONS · p. 18, Figure 6 · p. 19

Informative sources

  • SPDAlign is generally more robust than RCT as rotations increase, particularly when sufficient informative source structure is available.Performance declines with more severe rotations across settings.Source: C SIMULATIONS · p. 18, Figure 7 · p. 19

Label shifts

  • Under simulated class imbalance, UOT still provides performance gains despite label shifts, supporting its use when class imbalance is present.UOT relaxes the marginal constraints used by optimal transport.Source: C SIMULATIONS · p. 18, Figure 8 · p. 20

Data availability

  • With limited target data, SPDAlign's advantage over RCT is smaller, but it consistently appears as more target samples become available.The passage suggests alignment quality progressively improves as data accumulate in online settings.Source: C SIMULATIONS · p. 18, Figure 9 · p. 20

SNR

  • As SNR falls, all methods decline; SPDAlign helps at -3 dB and -10.5 dB and does not degrade results at -14.6 dB and -17.4 dB.The figure characterizes these as mild, moderate, and more extreme noise conditions, respectively.Source: C SIMULATIONS · p. 18, Figure 10 · p. 20

Domain numbers

  • Without source-domain rotation correction, training performance degrades as source domains increase because conflicting domain-specific rotations accumulate.This highlights source-domain alignment in multi-source settings.Source: C SIMULATIONS · p. 18, Figure 11 · p. 20

D.1 DATASET DETAILS

The dataset section covers motor imagery, workload, mental stress, and driving fatigue EEG, with distinct recording paradigms and preprocessing choices.

Motor imagery

  • The motor imagery dataset includes nine subjects, 22 channels, two sessions, and left- versus right-hand imagery trials.Signals were resampled to 256 Hz, filtered from 4–36 Hz, and segmented into 3-second epochs for the two most discriminative labels.Source: §D.1 · p. 21

Workload

  • The workload dataset includes 15 subjects performing low-, medium-, and high-workload MATB-II tasks across three sessions.Signals were filtered from 4–36 Hz, resampled to 200 Hz, and segmented into 2-second epochs.Source: §D.1 · p. 21

Mental Stress

  • The mental-stress dataset records resting and arithmetic-induced stress states from 36 subjects in one session.Preprocessing applies a 30 Hz low-pass filter and 50 Hz notch filter, resamples to 500 Hz, and segments 4-second epochs.Source: §D.1 · p. 21

Driving Fatigue

  • The driving-fatigue dataset uses a 90-minute driving simulation with 11 subjects and monitors lane departures and reaction times.Signals were filtered from 1–50 Hz, resampled to 128 Hz, and segmented into 3-second epochs.Source: §D.1 · p. 21

D.2 IMPLEMENTATION DETAILS

The implementation uses standard training and validation procedures with separate Euclidean and Riemannian optimizers; its feature extractor combines linear and covariance-based layers.

D.2 IMPLEMENTATION DETAILS

  • Each session is treated as a domain, and evaluation uses leave-one-subject-out or 10-fold cross-validation with subjects grouped for cross-subject adaptation.Source data are split 80/20 for training and validation, then trained for 50 epochs with cross-entropy loss.Source: §D.2 · p. 21
  • Euclidean and geometric frameworks use Adam and Riemannian Adam, respectively, with default PyTorch hyperparameters; models use official repositories and default hyperparameters.Experiments ran on standard PCs with 32-core CPUs, 128 GB of RAM, and one GPU.Source: §D.2 · p. 21

E.1 MODEL PARAMETERS

  • The feature extractor combines two linear convolutional layers, covariance pooling, and BiMap and ReEig layers.BiMap performs a linear subspace projection; ReEig thresholds eigenvalues at 10^-4, a threshold that was never activated.Source: §E.1 · p. 22

E.2 SPDALIGN SOURCE DOMAIN TRAINING

Source-domain training extracts SPD features, tracks domain-specific running Fréchet means, and applies a rotation-alignment stage before classifier fine-tuning. The appendix also describes an iterative Wasserstein Procrustes procedure and identifies an interpretability figure for a cross-subject workload setting.

E.2 SPDALIGN SOURCE DOMAIN TRAINING

  • Source training extracts SPD features, updates domain-specific running Fréchet means, then aligns source-domain rotations using class-constrained correspondences before fine-tuning the classifier.Unequal class-wise sample counts are downsampled; the feature extractor is frozen during rotation alignment and classifier fine-tuning.Source: §E.2 · p. 22

E.3 WASSERSTEIN PROCRUSTES IN SPDALIGN

  • Wasserstein Procrustes initializes an orthogonal matrix, iteratively computes transport couplings and gradients, and projects updates onto the orthogonal group using an SVD.The algorithm uses uniform weights for source and target samples.Source: §E.3 · p. 23

E.4 INTERPRETABILITY

  • Figure 12 presents extracted patterns from the workload dataset for class high in the cross-subject setting.Source: Figure 12 · p. 23

Related papers

Loading 2610.06315v1…