RepSPD: Enhancing SPD Manifold Representation in EEGs via Dynamic Graphs

Haohui Jia; Zheng Chen; Lingwei Zhu; Xu Cao; Yasuko Matsubara; Takashi Matsubara; Yasushi Sakurai

cs.AIFeb 26, 2026

RepSPD: Enhancing SPD Manifold Representation in EEGs via Dynamic Graphs

Authors: Haohui Jia, Zheng Chen, Lingwei Zhu, Xu Cao, Yasuko Matsubara, Takashi Matsubara, Yasushi Sakurai

Organizations: Information Science and Techinology, Hokkaido University, Japan · SANKEN, The University of Osaka, Japan · Great Bay University, China · Department of Computer Science, University of Illinois Urbana-Champaign, USA

Abstract

Decoding brain activity from electroencephalography (EEG) is crucial for neuroscience and clinical applications. Among recent advances in deep learning for EEG, geometric learning stands out as its theoretical underpinnings on symmetric positive definite (SPD) allows revealing structural connectivity analysis in a physics-grounded manner. However, current SPD-based methods focus predominantly on statistical aggregation of EEGs, with frequency-specific synchronization and local topological structures of brain regions neglected. Given this, we propose RepSPD, a novel geometric deep learning (GDL)-based model. RepSPD implements a cross-attention mechanism on the Riemannian manifold to modulate the geometric attributes of SPD with graph-derived functional connectivity features. On top of this, we introduce a global bidirectional alignment strategy to reshape tangent-space embeddings, mitigating geometric distortions caused by curvature and thereby enhancing geometric consistency. Extensive experiments demonstrate that our proposed framework significantly outperforms existing EEG representation methods, exhibiting superior robustness and generalization capabilities.

Figures & tables

Figure 1: Concept of SPD point projections from the manifold to tangent space. Our goal is to modulate SPD representations to achieve effective tangent embeddings.

Figure 2: Overview of the proposed RepSPD . Raw EEG signals and graph-constructed functional connectivity are separately encoded into SPD matrices. These SPD representations are processed via SPDNet to generate query, key, and value matrices for modulating the SPD with cross-attention. Our dynamic graph guided modulation operates on the Riemannian SPD manifold, computing attention weights based on Log-Euclidean distance and modulating the EEG representation with graph structures. A structure-aware loss function, combining task loss and a geometry-level alignment loss $\mathcal{L}_{GeoTop}$ , ensures functional-structural consistency and robust discriminative representations in tangent space.

Figure 3: Accuracy comparison on seizure detection and MI full classification. Bars show the mean accuracy and error bars indicate the standard deviation across 5-fold runs.

Figure 4: Visualization of tangent-space with 2D PCA. (a) RepSPD produces well-structured clusters. (b) w/o $\mathcal{L}_{GeoTop}$ leads to increased overlap. (c) MAtt captures spatiotemporal patterns via manifold attention but yields looser clusters. (d) SPDNet exhibits the mixing clusters by the second-order statistics.

Figure 5: Visualization of topological map of brain across different methods. Compared to SPDNet and RepSPD w/o proposed $\mathcal{L}_{GeoTop}$ , our RepSPD more accurately restores the functional spatial pattern observed in the ground truth.

Figure 6: Gradient-level visualization of attention maps. (a) Spectrogram of input EEGs for each class. (b) RepSPD produces more structured and dynamic attention patterns that emphasize time–frequency regions. (c) MAtt yields less class-specific attention.

Query input	ACC	AUROC	F1
edge SPD	86.13	81.66	41.88
node SPD	86.43	81.61	42.92
edge and node mixed SPD	85.17	80.83	41.44
node and $S$ mixed SPD	85.22	80.47	40.42

Table 1: Ablation comparison with different query options, including edge-based, node-based, node and edge mixed-based, and node-EEG SPD mixed queries.

$\lambda$	ACC	AUROC	F1
0 (w/o $\mathcal{L}_{\text{GeoTop}}$ )	77.81	79.57	37.80
0.1	80.52	76.61	36.90
0.3	86.43	81.61	42.92
0.5	82.31	81.47	39.12
0.7	76.87	79.13	38.92

Table 2: Ablation comparison with geometry-aware feature space alignment loss

\mathcal{L}_{\text{GeoTop}}

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Dynamic graph embedding-based SPD modulation via the proposed manifold cross-attention. Given the estimated SPD matrix $s_{i}$ and graph-constructed SPD matrix $c_{i}$ , BiMap projections yield query $(q_{i})$ , key $(k_{i})$ , and value $(v_{i})$ representations. A Log-Euclidean metric computes attention scores $d_{\mathrm{LE}}(q_{i},k_{i})$ , which modulate the value embedding in SPD space through $\mathbb{E}[\log(v_{i})\mathcal{C}_{\mathcal{A}}]$ , enabling geometry-aware feature integration.

Method	Type	ACC	AUROC	F1
QR-projector	Manifold	83.5	76.3	35.5
QR-projector w/o $\mathcal{L}_{\text{GeoTop}}$	Manifold	79.0	74.7	33.5
SVD-projector	Manifold	79.6	77.2	34.1
SVD-projector w/o $\mathcal{L}_{\text{GeoTop}}$	Manifold	72.6	73.8	30.3
cross-attention	Euclidean	84.0	79.6	38.4
RepSPD	Manifold	86.4 ( $\uparrow 2.4$ )	81.6 ( $\uparrow 2.0$ )	42.9 ( $\uparrow 4.5$ )

Appendix

Table 3: Ablation comparison on TUSZ dataset for 12s seizure detection with QR, SVD-projector, and regular cross-attention. Bold and underline indicate best best results.

Figure 8: Topographic visualizations of spatial brain activation at selected time segments ( $T_{1}$ , $T_{5}$ , $T_{9}$ , $T_{12}$ ) from one EEG trial. Each map illustrates the activation over 19 channels projected to 2D scalp topography. We compare the output of our method ( RepSPD ), the ablated variant without geometric alignment loss ( $\mathcal{L}_{\text{GeoTop}}$ ), and the baseline SPDNet against the ground truth.

Method	Mode	ACC	AUROC	F1
RepSPD w	Train	89.4	81.6	45.2
RepSPD w	Test	86.4	81.6	42.9
RepSPD w/o	Train	88.4	83.0	43.4
RepSPD w/o	Test	78.8( $\downarrow 9.6$ )	73.6 ( $\downarrow 9.4$ )	34.8 ( $\downarrow 8.6$ )

Appendix

Table 4: Ablation comparison with meta optimization.

Top- $\tau$	AUROC	ACC	F1
2	0.807 $\pm$ 0.003	0.815 $\pm$ 0.003	0.404 $\pm$ 0.009
3	0.816 $\pm$ 0.006	0.864 $\pm$ 0.003	0.429 $\pm$ 0.017
7	0.809 $\pm$ 0.004	0.862 $\pm$ 0.004	0.411 $\pm$ 0.013
9	0.801 $\pm$ 0.004	0.849 $\pm$ 0.004	0.417 $\pm$ 0.011
11	0.776 $\pm$ 0.004	0.821 $\pm$ 0.002	0.419 $\pm$ 0.003
13	0.805 $\pm$ 0.003	0.832 $\pm$ 0.004	0.414 $\pm$ 0.003

Appendix

Table 5: Ablation on TUSZ dataset for 12s seizure detection with different top-

\tau

options. Bold and underline indicate best and second-best results.

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SPERA: Spherical Prior EEG Foundation Model with Geometry- and Frequency-Aware Latent Prediction

Minsu Kim, Ye-Sung Kim, Hyeseong Jeon, Wooseok Hyung+2

Department of Electronic Engineering, Hanyang University, Seoul, Republic of Korea · Department of Artificial Intelligence, Hanyang University, Seoul, Republic of Korea · Department of Biomedical Engineering, Hanyang University, Seoul, Republic of Korea

EEG Foundation Models Predictive Representation Learning Self-Supervised Time Series Representation Learning

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Electroencephalography (EEG) provides a non-invasive measure of ongoing neural activity, but building general-purpose EEG models remains challenging due to the heterogeneity of subjects, devices, and electrode montages. Existing EEG foundation models predominantly rely on reconstruction-based objectives defined on the observed signal, which contains both neural and non-neural components. We introduce SPERA (Spherical Prior EEG Representation Architecture), an EEG foundation model that adopts the joint-embedding predictive architecture (JEPA) to predict in latent space. SPERA introduces a Legendre-polynomial spatial prior, incorporated into attention to encode varying scalp electrode geometries. Two further components adapt the model to EEG: factorized temporal and spatial attention interleaved with periodic full-attention blocks, and a relational spectral regularizer aligning latent similarity structure with spectral views. Pretrained on approximately 80,000 hours of EEG from 29,048 subjects across 106 datasets, SPERA achieves the highest average balanced accuracy across nine downstream tasks spanning clinical, cognitive, and BCI applications. SPERA further exhibits strong parameter efficiency under linear probing and robustness across varying recording conditions, suggesting its potential as a general-purpose backbone for diverse EEG analyses.

Abstract

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