HEEGNet: Hyperbolic Embeddings for EEG

AI summary

Overview

Research area: Machine learning for neuroscience — specifically, geometric deep learning (hyperbolic neural networks) applied to electroencephalography (EEG) decoding and cross-domain generalization.

Technical level: Advanced. The paper assumes familiarity with Riemannian geometry, the Lorentz model of hyperbolic space, gyrovector operations, optimal transport (sliced-Wasserstein distances), and batch normalization techniques for domain adaptation.

Scope: The paper argues that EEG representations are inherently hierarchical and therefore better embedded in hyperbolic rather than Euclidean space, and proposes HEEGNet — a hybrid Euclidean/hyperbolic network with a two-stage "coarse-to-fine" domain alignment strategy — evaluated across visual evoked potential, emotion recognition, and intracranial EEG datasets.

What This Paper Is About

EEG-based brain-computer interfaces work poorly when deployed on new subjects or new recording sessions, because the statistical distribution of EEG signals shifts from domain to domain. The authors' core claim is that EEG carries hierarchical structure (as suggested by prior work on visual processing, emotion regulation, intracranial and scalp EEG), and that hyperbolic space — which grows exponentially with radius, matching tree-like hierarchies — is a more natural geometry for EEG embeddings than flat Euclidean space. The goal is a single architecture that both captures that hierarchy and produces domain-invariant embeddings that transfer across unseen, unlabeled target domains.

Key Contributions

1. Empirical evidence of hyperbolicity in EEG. A pilot study uses EEGNet to generate embeddings for five datasets and quantifies their tree-likeness with the scale-invariant δ-hyperbolicity measure δ_rel, on raw EEG, intermediate-layer embeddings, and the classification space. All datasets show hierarchical structure.
2. A pilot demonstration that hyperbolic classifiers generalize better. Replacing EEGNet's Euclidean multinomial logistic regression (MLR) with a hyperbolic MLR (HMLR) improves balanced accuracy on all five pilot datasets.
3. HEEGNet, a hybrid hyperbolic architecture. A shared feature extractor f_θ built from three Euclidean convolutional layers (taken from EEGNet), a projection layer (ProjX) into the Lorentz model, and a hyperbolic convolutional layer; a domain-specific alignment module m_φ; and a hyperbolic MLR classifier g_ψ. The model is trained end-to-end with a cross-entropy loss plus the HHSW term.
4. DSMDBN, a coarse-to-fine domain adaptation strategy. Domain-specific moment-then-distribution batch normalization first aligns domain-specific first- and second-order moments in hyperbolic space using Riemannian batch normalization, then minimizes the Hyperbolic Horospherical Sliced-Wasserstein (HHSW) discrepancy between each source domain distribution and a standard hyperbolic Gaussian.

Main Findings

• All five pilot datasets are close to hyperbolic. Reported δ_rel (lower means more hyperbolic), for Nakanishi (n=9), Wang (n=34), Seed (n=45), Faced (n=123), and Boran (n=37): raw EEG 0.244±0.064, 0.219±0.053, 0.052±0.026, 0.103±0.077, 0.157±0.045; intermediate embeddings 0.263±0.036, 0.240±0.045, 0.061±0.026, 0.111±0.050, 0.141±0.041; classification space 0.306±0.027, 0.333±0.039, 0.072±0.025, 0.132±0.047, 0.017±0.048.
• Hyperbolic classifiers beat Euclidean ones in the pilot. EEGNet + HMLR versus EEGNet (balanced accuracy): Nakanishi 60.8±20.7 vs 57.2±19.8; Wang 39.2±13.8 vs 37.4±12.9; Seed 75.1±20.0 vs 74.5±19.8; Faced 38.8±12.4 vs 24.8±13.7; Boran 57.4±8.7 vs 55.4±9.1.
• HEEGNet is state of the art on the grand average. HEEGNet (DSMDBN) reaches VEP 79.6±19.8, emotion 82.8±16.6, intracranial 58.4±9.6, overall 78.7±18.6. HEEGNet (DSMDBN+EA) reaches VEP 77.4±12.6, emotion 86.7±12.5, intracranial 66.7±12.5, overall 81.7±14.4.
• The strongest baselines fall well short on the overall average. For comparison, EEGConformer + EA reaches 70.9±24.1 overall (66.1±27.2 without EA), EEGNet + EA 67.4±22.3 (39.9±23.6 without EA), and SchirrmeisterNet + EA 66.5±25.7 (49.7±23.1 without EA).
• Moment alignment in hyperbolic space is the primary driver of gains. In the ablation, the best configuration combined moment, distribution, and input alignment; relative to it, removing input alignment cost 3.0 points, removing distribution alignment cost 5.0 points, and removing moment alignment cost at least 12.3 points (other moment-free configurations lost 12.9 and 15.5 points). Distribution alignment only helped when paired with moment alignment.
• Latent-space moment alignment outperforms input-space alignment, consistent with prior findings the authors cite, and combining DSMDBN with input-space alignment (EA) gave the best overall result.
• Qualitative separability improves. t-SNE visualizations for Nakanishi (subject 4, session 1) show that hyperbolic embedding and the two DSMDBN stages progressively improve class separability.
• Motor imagery, a non-hierarchical modality, was also tested. The paper states it evaluates HEEGNet on public motor imagery datasets resampled to 250 or 256 Hz with 4–36 Hz temporal filters and 3-second epochs, but the provided content is truncated before those results are reported.

Methodology in Plain English

The authors start from a hypothesis rather than an architecture: if EEG encodes hierarchical brain processes, then the geometry of the representation space should be hyperbolic, not flat. They test this cheaply first by training a standard EEGNet and measuring how tree-like its internal embeddings are, then by swapping only the final classifier for a hyperbolic one. Both tests support the hypothesis.

Building on that, HEEGNet keeps the parts of EEG decoding that already work well — ordinary convolutional layers that extract frequency, electrode, and time patterns from raw EEG — and then projects those features into hyperbolic space, where a hyperbolic convolution and classifier refine them. Keeping the front end Euclidean is deliberate: convolutional EEG operations carry known neurophysiological meaning, and fully hyperbolic operations do not preserve those physical properties (the authors report fully hyperbolic experiments separately).

To handle the domain-shift problem, the paper extends an existing state-of-the-art recipe (domain-specific momentum batch normalization) with a second stage. Stage one normalizes each domain's features using its own running estimates of the mean and variance, computed with hyperbolic (Fréchet mean / variance) operations. Stage two pushes each source domain's feature distribution toward a common standard hyperbolic Gaussian by minimizing the hyperbolic horospherical sliced-Wasserstein discrepancy — a hyperbolic analogue of the sliced-Wasserstein distance, obtained by projecting distributions onto horospheres. This is "coarse-to-fine": you cannot align distributions that are too far apart, so moments are aligned first to bring them into range. At test time only stage one is applied, since target labels and source data are unavailable; the source-trained, Gaussian-aligned feature extractor is sufficient to normalize target features.

Why This Matters

Impact on research: The paper reframes EEG domain adaptation as a geometry problem rather than purely a training-recipe problem, and it supplies a concrete, reusable recipe (a hyperbolicity diagnostic followed by a hyperbolic architecture plus two-stage alignment) plus public code. It also connects two previously separate EEG lines — the neuroscience literature on hierarchical brain processing and the machine learning literature on hyperbolic neural networks — and claims state-of-the-art results across three EEG modalities spanning 272 evaluated domains in total.

Real-world applications (as framed by the paper):

• Steady-state visually evoked potential (SSVEP) and code-modulated VEP spellers/BCIs, where models must work on new users without lengthy calibration.
• Emotion recognition systems, which the paper situates in human-computer interaction and healthcare.
• Intracranial EEG analysis for clinical and cognitive neuroscience research, where signals are recorded across electrodes that differ between subjects.
• Attention monitoring and broader assistive/rehabilitation neurotechnology, which the authors cite as motivating applications.

Industry relevance: The paper targets settings where labeled calibration data are expensive, unavailable, or cannot be shared for privacy reasons. Its source-free, multi-source multi-target formulation means a model can be shipped and adapted to a new user without access to other users' raw data — a practical requirement for consumer and clinical BCI products, and for the alignment methods (EA, STMA) that are model-agnostic and can be bolted onto existing pipelines.

Future Directions

• Fully hyperbolic architectures. The paper notes that its hybrid design is chosen partly because Euclidean operations do not preserve physical properties in hyperbolic space, and points to separate fully hyperbolic experiments in an appendix — leaving open how much can be moved into hyperbolic space without losing neurophysiological fidelity.
• Motor imagery and non-hierarchical modalities. The authors explicitly evaluate on motor imagery datasets "not reported to encode hierarchical information," which tests whether the hyperbolic advantage is specific to hierarchical tasks or more general. Those results are not reported in the available content.
• Tuning the alignment trade-off. The HHSW loss weight was set to 0.01 for emotion and 0.5 for all other datasets and curvature was fixed at -1 by default, suggesting sensitivity analysis across datasets, curvatures, and other hyperbolic models remains open.
• Scaling and cost. The paper discusses the computational cost of hyperbolic operations separately, and the DSMBDN approach depends on estimating per-domain statistics under Riemannian Adam with 100 epochs and 80/20 source splits — questions remain about behavior with many more domains, larger datasets, and clinical deployment constraints.

Target Audience

Researchers and graduate students working on EEG decoding, brain-computer interfaces, or transfer learning under distribution shift will get the most from this paper, particularly those already comfortable with Riemannian or hyperbolic geometry. It is also relevant to machine learning researchers interested in geometric deep learning applied to biosignals, and to practitioners building BCIs who need source-free or privacy-preserving adaptation methods. Readers without a background in manifold-valued optimization will find the preliminaries section (Lorentz model, Fréchet means, gyrovector operations, sliced-Wasserstein) necessary but demanding.

Authors’ abstract

Electroencephalography (EEG)-based brain-computer interfaces facilitate direct communication with a computer, enabling promising applications in human-computer interactions. However, their utility is currently limited because EEG decoding often suffers from poor generalization due to distribution shifts across domains (e.g., subjects). Learning robust representations that capture underlying task-relevant information would mitigate these shifts and improve generalization. One promising approach is to exploit the underlying hierarchical structure in EEG, as recent studies suggest that hierarchical cognitive processes, such as visual processing, can be encoded in EEG. While many decoding methods still rely on Euclidean embeddings, recent work has begun exploring hyperbolic geometry for EEG. Hyperbolic spaces, regarded as the continuous analogue of tree structures, provide a natural geometry for representing hierarchical data. In this study, we first empirically demonstrate that EEG data exhibit hyperbolicity and show that hyperbolic embeddings improve generalization. Motivated by these findings, we propose HEEGNet, a hybrid hyperbolic network architecture to capture the hierarchical structure in EEG and learn domain-invariant hyperbolic embeddings. To this end, HEEGNet combines both Euclidean and hyperbolic encoders and employs a novel coarse-to-fine domain adaptation strategy. Extensive experiments on multiple public EEG datasets, covering visual evoked potentials, emotion recognition, and intracranial EEG, demonstrate that HEEGNet achieves state-of-the-art performance. The code is available at https://github.com/fightlesliefigt/HEEGNet