A Unified SPD Token Transformer Framework for EEG Classification: Systematic Comparison of Geometric Embeddings
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908796551430144 |
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| author | Chen, Chi-Sheng Kuo, En-Jui Chen, Guan-Ying Zhang, Xinyu Zhang, Fan |
| author_facet | Chen, Chi-Sheng Kuo, En-Jui Chen, Guan-Ying Zhang, Xinyu Zhang, Fan |
| contents | Spatial covariance matrices of EEG signals are Symmetric Positive Definite (SPD) and lie on a Riemannian manifold, yet the theoretical connection between embedding geometry and optimization dynamics remains unexplored. We provide a formal analysis linking embedding choice to gradient conditioning and numerical stability for SPD manifolds, establishing three theoretical results: (1) BWSPD's $\sqrtκ$ gradient conditioning (vs $κ$ for Log-Euclidean) via Daleckii-Kre\uın matrices provides better gradient conditioning on high-dimensional inputs ($d \geq 22$), with this advantage reducing on low-dimensional inputs ($d \leq 8$) where eigendecomposition overhead dominates; (2) Embedding-Space Batch Normalization (BN-Embed) approximates Riemannian normalization up to $O(\varepsilon^2)$ error, yielding $+26\%$ accuracy on 56-channel ERP data but negligible effect on 8-channel SSVEP data, matching the channel-count-dependent prediction; (3) bi-Lipschitz bounds prove BWSPD tokens preserve manifold distances with distortion governed solely by the condition ratio $κ$. We validate these predictions via a unified Transformer framework comparing BWSPD, Log-Euclidean, and Euclidean embeddings within identical architecture across 1,500+ runs on three EEG paradigms (motor imagery, ERP, SSVEP; 36 subjects). Our Log-Euclidean Transformer achieves state-of-the-art performance on all datasets, substantially outperforming classical Riemannian classifiers and recent SPD baselines, while BWSPD offers competitive accuracy with similar training time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21521 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Unified SPD Token Transformer Framework for EEG Classification: Systematic Comparison of Geometric Embeddings Chen, Chi-Sheng Kuo, En-Jui Chen, Guan-Ying Zhang, Xinyu Zhang, Fan Machine Learning Human-Computer Interaction Spatial covariance matrices of EEG signals are Symmetric Positive Definite (SPD) and lie on a Riemannian manifold, yet the theoretical connection between embedding geometry and optimization dynamics remains unexplored. We provide a formal analysis linking embedding choice to gradient conditioning and numerical stability for SPD manifolds, establishing three theoretical results: (1) BWSPD's $\sqrtκ$ gradient conditioning (vs $κ$ for Log-Euclidean) via Daleckii-Kre\uın matrices provides better gradient conditioning on high-dimensional inputs ($d \geq 22$), with this advantage reducing on low-dimensional inputs ($d \leq 8$) where eigendecomposition overhead dominates; (2) Embedding-Space Batch Normalization (BN-Embed) approximates Riemannian normalization up to $O(\varepsilon^2)$ error, yielding $+26\%$ accuracy on 56-channel ERP data but negligible effect on 8-channel SSVEP data, matching the channel-count-dependent prediction; (3) bi-Lipschitz bounds prove BWSPD tokens preserve manifold distances with distortion governed solely by the condition ratio $κ$. We validate these predictions via a unified Transformer framework comparing BWSPD, Log-Euclidean, and Euclidean embeddings within identical architecture across 1,500+ runs on three EEG paradigms (motor imagery, ERP, SSVEP; 36 subjects). Our Log-Euclidean Transformer achieves state-of-the-art performance on all datasets, substantially outperforming classical Riemannian classifiers and recent SPD baselines, while BWSPD offers competitive accuracy with similar training time. |
| title | A Unified SPD Token Transformer Framework for EEG Classification: Systematic Comparison of Geometric Embeddings |
| topic | Machine Learning Human-Computer Interaction |
| url | https://arxiv.org/abs/2601.21521 |