HEEGNet: Hyperbolic Embeddings for EEG

Conference: ICLR 2026
arXiv: 2601.03322
Code: GitHub
Area: Brain-Computer Interface/Geometric Deep Learning
Keywords: EEG, Hyperbolic Space, Domain Adaptation, Hierarchical Structure, BCI

TL;DR

This paper provides the first systematic validation of the hyperbolic nature (hierarchical structure) of EEG data. It proposes HEEGNet, a hybrid Euclidean-hyperbolic network architecture that combines a Euclidean encoder for spatio-temporal-spectral feature extraction with a hyperbolic encoder to capture hierarchical relationships. Coupled with a novel coarse-to-fine domain adaptation strategy (DSMDBN), it achieves SOTA results across cross-domain tasks in visual evoked potentials, emotion recognition, and intracranial EEG.

Background & Motivation

Background: EEG-based BCIs suffer from poor generalization due to inter-subject/inter-session distribution shifts. Domain Adaptation (DA) methods (such as moment alignment) are current SOTA but fail under large shifts. EEG decoding has been almost exclusively based on Euclidean embeddings.

Limitations of Prior Work: (1) Cognitive processes such as visual processing and emotion regulation in the brain possess hierarchical structures, which are difficult to represent efficiently in Euclidean space—where the circumference of a circle grows linearly while the number of tree nodes grows exponentially; (2) Moment alignment alone cannot guarantee positive transfer, especially under large domain shifts.

Key Challenge: The hierarchical structure of EEG features requires exponential representational capacity, which Euclidean space lacks, yet hyperbolic neural networks have not been systematically explored for EEG.

Key Insight: Preliminary research shows that EEG exhibits hyperbolicity (low \(\delta_{rel}\)). Replacing Euclidean MLR with hyperbolic MLR alone improves cross-domain performance, proving that hyperbolic embeddings indeed aid EEG generalization.

Core Idea: Utilizing hyperbolic space to capture the hierarchical structure of EEG combined with two-stage domain adaptation (moment alignment → distribution alignment) to achieve cross-domain generalization.

Method

Overall Architecture

HEEGNet addresses EEG cross-domain generalization by capturing the natural hierarchical structure of cognitive processes while mitigating distribution shifts across subjects/sessions. It decouples signal processing from geometric representation: a Euclidean encoder (temporal → spatial → temporal convolutions) extracts spectral-spatial-temporal features from raw EEG, which are then projected via ProjX into the Lorentz hyperbolic space. Hyperbolic convolutions are used to refine hierarchical relationships, with DSMDBN two-stage domain adaptation interspersed to eliminate domain shifts. Finally, a hyperbolic MLR classifier outputs predictions.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400, 'subGraphTitleMargin': {'top': 8, 'bottom': 16}}}}%%
flowchart TD
    A["Raw EEG Signals"] --> SG1
    subgraph SG1["Hybrid Euclidean-Hyperbolic Architecture (Design 1)"]
        direction TB
        B["Euclidean Encoder<br/>T→S→T Convolutions<br/>Extract Spectral-Spatial-Temporal Features"] --> C["ProjX Mapping to<br/>Lorentz Hyperbolic Space"]
        C --> D["Hyperbolic Pointwise Conv<br/>Refine Hierarchical Relations"]
    end
    SG1 --> SG2
    subgraph SG2["DSMDBN Two-Stage Domain Adaptation (Design 2)"]
        direction TB
        E["DSMDBN(1) Moment Alignment<br/>Gyro-subtraction/multiplication for centering+scaling"] --> F["DSMDBN(2) Distribution Alignment<br/>Minimize HHSW Divergence<br/>Align to Standard Hyperbolic Gaussian"]
    end
    SG2 --> G["Hyperbolic MLR Classifier<br/>Scoring via Point-to-Hyperplane Distance"]
    G --> H["Cross-Domain Prediction"]

Key Designs

1. Hybrid Euclidean-Hyperbolic Architecture: Decoupling Signal Processing and Hierarchical Representation

Pure hyperbolic networks discard valuable signal processing priors in EEG decoding, while pure Euclidean networks cannot represent hierarchical structures (Euclidean space cannot accommodate exponential hierarchies). HEEGNet uses a two-stage approach: the front end consists of three EEGNet-style convolutional layers for spectral-spatial-temporal feature extraction with neurophysiological interpretability; then, ProjX projects features into the Lorentz model \(\mathbb{L}_K^n\) for pointwise convolutions. The front end transforms signals into meaningful features, while the back end organizes hierarchical relationships in a space with exponential representational capacity.

2. DSMDBN: Coarse-to-Fine Two-Stage Domain Adaptation

Moment alignment (current SOTA) fails under large domain shifts. DSMDBN adds distribution alignment atop moment alignment. Stage one, DSMDBN(1), uses Riemannian Batch Normalization for domain-specific moment alignment—centering is implemented via gyro-subtraction and scaling via gyro-multiplication in hyperbolic space, aligning the mean and scale of each domain. Stage two, DSMDBN(2), minimizes the HHSW divergence to align the distribution of each source domain to a standard hyperbolic Gaussian \(\mathcal{N}(\bar{0}, 1)\). This "coarse-to-fine" approach aligns first-order/second-order statistics before providing stronger theoretical guarantees for distribution alignment.

3. Hyperbolic Operators on the Lorentz Model: Geometric Toolbox

Hyperbolic operations are built on the gyro-algebra of the Lorentz model: gyroaddition, gyromultiplication, and gyroinverse replace Euclidean operations. Fréchet mean and variance are redefined on the Lorentz model (supporting DSMDBN centering/scaling), and hyperbolic MLR uses hyperbolic distance from points to hyperplanes for classification. Notably, replacing only the Euclidean MLR head with a hyperbolic MLR improves cross-domain performance across all datasets, indicating this geometric toolset is inherently better suited for the hierarchical nature of EEG.

Loss & Training

• Total Loss = Classification Loss + HHSW Distribution Alignment Loss.
• Domain-specific momentum batch normalization: Statistics are updated with decaying momentum during training and fixed during testing.
• Riemannian Adam optimizer is used for parameter updates on the manifold.

Key Experimental Results

Background

Dataset	\(\delta_{rel}\) Raw EEG	\(\delta_{rel}\) Embedding Layer	Description
Nakanishi	Low	Low	Visual
Wang	Low	Low	Visual
Seed	Low	Low	Emotional
Faced	Low	Low	Emotional
Boran	Low	Low	Intracranial

→ All datasets exhibit low \(\delta_{rel}\), confirming the hyperbolicity of EEG.

Main Results

Cross-subject/Cross-session Adaptation:

Method	Visual EEG	Emotional EEG	Intracranial EEG	Average
EEGNet	Baseline	Baseline	Baseline	Baseline
EEGNet+HMLR	↑	↑	↑	Stable Gain
HEEGNet	SOTA	SOTA	SOTA	Overall Best

Key Findings

• Replacing MLR with hyperbolic MLR alone improves performance across all datasets → hyperbolic geometry is better suited for EEG.
• t-SNE visualization shows hyperbolic embeddings have significantly better class separability than Euclidean ones.
• The two-stage DSMDBN strategy significantly outperforms moment alignment alone.
• Performance gains are also observed on motor imagery datasets (where hierarchy was not explicitly reported) → suggests possible unidentified hierarchical structures.

Highlights & Insights

• First systematic validation of EEG hyperbolicity: Quantitative analysis via \(\delta_{rel}\) across multiple datasets provides a solid empirical foundation for this direction.
• Rationality of Hybrid Architecture: Instead of pure hyperbolic layers (which lose signal processing priors), HEEGNet extracts features in Euclidean space before mapping to hyperbolic space—a design philosophy applicable to other domains.
• Coarse-to-fine DSMDBN: Mapping moment alignment → distribution alignment is a natural two-step strategy, where the former aligns mean and scale, and the latter aligns the overall distribution shape.

Limitations & Future Work

• Hyperbolic operations have higher computational overhead (exp/log maps) than Euclidean ones, affecting real-time BCI.
• Curvature \(K\) is a hyperparameter requiring tuning; adaptive curvature learning may be better.
• HHSW might require many projection directions for accuracy in high dimensions.
• Electrode counts vary across subjects in intracranial EEG, limiting cross-subject experimental setups.

Related Work & Insights

• vs. EEGNet: HEEGNet enhances EEGNet with hyperbolic layers and DSMDBN for across-the-board improvements.
• vs. Chang et al. (Hyperbolic EEG): They focus on contrastive pre-training; HEEGNet designs a complete architecture and domain adaptation scheme.
• vs. Riemannian methods like SPDNet: SPDNet operates on covariance matrix manifolds, while HEEGNet operates in hyperbolic space, focusing on different geometric structures.

Rating

• Novelty: ⭐⭐⭐⭐⭐ First systematic proof of EEG hyperbolicity and hybrid hyperbolic architecture.
• Experimental Thoroughness: ⭐⭐⭐⭐⭐ From background studies to multi-dataset, multi-task, and comprehensive ablation studies.
• Writing Quality: ⭐⭐⭐⭐ Sufficient background introduction and clear methodological description.
• Value: ⭐⭐⭐⭐ Introduces a new geometric perspective to EEG decoding and opens the hyperbolic EEG direction.

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