# A Unified SPD Token Transformer Framework for EEG Classification: Systematic Comparison of Geometric Embeddings

arXiv: https://arxiv.org/abs/2601.21521 · HTML: https://mlsanity.com/paper/2601.21521/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 11 of 11)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces](https://mlsanity.com/paper/1611.08024/) (2016)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [EEG-TCNet: An Accurate Temporal Convolutional Network for Embedded Motor-Imagery Brain-Machine Interfaces](https://mlsanity.com/paper/2006.00622/) (2020)
- [FBCNet: A Multi-view Convolutional Neural Network for Brain-Computer Interface](https://mlsanity.com/paper/2104.01233/) (2021)
- [MAtt: A Manifold Attention Network for EEG Decoding](https://mlsanity.com/paper/2210.01986/) (2022)
- [Mind's Eye: Image Recognition by EEG via Multimodal Similarity-Keeping Contrastive Learning](https://mlsanity.com/paper/2406.16910/) (2024)
- [NECOMIMI: Neural-Cognitive Multimodal EEG-informed Image Generation with Diffusion Models](https://mlsanity.com/paper/2410.00712/) (2024)
- [Structure-Preserving Transformers for Sequences of SPD Matrices](https://mlsanity.com/paper/2309.07579/) (2023)

## Abstract (arXiv)

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 √kappa gradient conditioning (vs kappa for Log-Euclidean) via Daleckii-Kreuin matrices provides better gradient conditioning on high-dimensional inputs (d ≥ 22), with this advantage reducing on low-dimensional inputs (d ≤ 8) where eigendecomposition overhead dominates; (2) Embedding-Space Batch Normalization (BN-Embed) approximates Riemannian normalization up to O(varepsilon2) 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 kappa. 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.