References#
This page provides an overview of the foundational literature behind SPD Learn, organized by model architecture and research lineage.
Literature Map#
The following visualization shows the relationships between key publications implemented in SPD Learn. The map illustrates how different SPD neural network architectures have evolved and influenced each other over time.
SPD Learn Literature Map — Generated with Litmaps. The x-axis represents publication date (more recent to the right), and the y-axis represents citation count (more cited at the top). Lines show citation relationships between papers.#
Model Legend#
The following legend explains the color coding used in the literature map above, with each color representing a distinct model family implemented in SPD Learn.
Full Bibliography#
Below are the complete bibliographic references for the models and foundational works implemented in SPD Learn.
Zhiwu Huang and Luc Van Gool. A riemannian network for spd matrix learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 31, 2036–2042. 2017. URL: https://ojs.aaai.org/index.php/AAAI/article/view/10866.
Ce Ju and Cuntai Guan. Tensor-cspnet: a novel geometric deep learning framework for motor imagery classification. IEEE Transactions on Neural Networks and Learning Systems, 34(12):10955–10969, 2023. doi:10.1109/TNNLS.2022.3172108.
Reinmar J Kobler, Jun-ichiro Hirayama, Qibin Zhao, and Motoaki Kawanabe. Spd domain-specific batch normalization to crack interpretable unsupervised domain adaptation in eeg. In Advances in Neural Information Processing Systems, volume 35, 6219–6235. 2022. URL: https://proceedings.neurips.cc/paper_files/paper/2022/hash/28ef7ee7cd3e03093acc39e1272411b7-Abstract-Conference.html.
Igor Carrara*, Bruno Aristimunha*, Marie-Constance Corsi, Raphael Y. de Camargo, Sylvain Chevallier, and Théodore Papadopoulo. Geometric neural network based on phase space for bci-eeg decoding. Journal of Neural Engineering, 21(6):016049, 2024. doi:10.1088/1741-2552/ad88a2.
Antoine Collas, Ce Ju, Nicolas Salvy, and Bertrand Thirion. Riemannian flow matching for brain connectivity matrices via pullback geometry. In The Thirty-ninth Annual Conference on Neural Information Processing Systems. 2025. URL: https://openreview.net/forum?id=NY3LzmUXl7.
Daniel Brooks, Olivier Schwander, Frédéric Barbaresco, Jean-Yves Schneider, and Matthieu Cord. Riemannian batch normalization for spd neural networks. In Advances in Neural Information Processing Systems, volume 32. 2019. URL: https://proceedings.neurips.cc/paper/2019/hash/6e69ebbfad976d4637bb4b39de261bf7-Abstract.html.
Nicholas J Higham. Accuracy and Stability of Numerical Algorithms. SIAM, 2nd edition, 2002. doi:10.1137/1.9780898718027.
Zhenhua Lin. Riemannian geometry of symmetric positive definite matrices via cholesky decomposition. SIAM Journal on Matrix Analysis and Applications, 40(4):1353–1370, 2019. doi:10.1137/18M1221084.
Ce Ju, Reinmar Kobler, Antoine Collas, Motoaki Kawanabe, Cuntai Guan, and Bertrand Thirion. Spd matrix learning for neuroimaging analysis: perspectives, methods, and challenges. arXiv preprint arXiv2504.18882, 2026. URL: https://arxiv.org/abs/2504.18882.
Mario Lezcano-Casado. Trivializations for gradient-based optimization on manifolds. In Advances in Neural Information Processing Systems, volume 32, 9154–9164. 2019. URL: https://proceedings.neurips.cc/paper/2019/hash/1b33d16fc562464579b7199ca3114982-Abstract.html.
Rajendra Bhatia. Positive Definite Matrices. Princeton University Press, 2007. doi:10.1515/9781400827787.
Marco Congedo, Alexandre Barachant, and Rajendra Bhatia. Riemannian geometry for eeg-based brain-computer interfaces; a primer and a review. Brain-Computer Interfaces, 4(3):155–174, 2017. doi:10.1080/2326263X.2017.1297192.
Xavier Pennec, Pierre Fillard, and Nicholas Ayache. A riemannian framework for tensor computing. International Journal of Computer Vision, 66(1):41–66, 2006. doi:10.1007/s11263-005-3222-z.
Tsuyoshi Ando, Chi-Kwong Li, and Roy Mathias. Geometric means. Linear Algebra and its Applications, 385:305–334, 2004. doi:10.1016/j.laa.2003.11.019.
Vincent Arsigny, Pierre Fillard, Xavier Pennec, and Nicholas Ayache. Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications, 29(1):328–347, 2007. doi:10.1137/050637996.
Rajendra Bhatia, Tanvi Jain, and Yongdo Lim. On the bures-wasserstein distance between positive definite matrices. Expositiones Mathematicae, 37(2):165–191, 2019. doi:10.1016/j.exmath.2018.01.002.
Paolo Zanini, Marco Congedo, Christian Jutten, Salem Said, and Yannick Berthoumieu. Transfer learning: a riemannian geometry framework with applications to brain–computer interfaces. IEEE Transactions on Biomedical Engineering, 65(5):1107–1116, 2017.
Ziheng Chen, Yue Song, Yunmei Xu, and Nicu Sebe. A lie group approach to riemannian batch normalization. In International Conference on Learning Representations. 2024. URL: https://openreview.net/forum?id=okYdj8Ysru.
Matthew Chen, Micky Nnamdi, Justin Shao, Andrew Hornback, Hongyun Huang, Ben Tamo, Yishan Zhong, Benoit Marteau, Wenqi Shi, and May Dongmei Wang. Mendr: manifold explainable neural data representations. 2025. arXiv:2508.04956.
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Isay Katsman, Eric Chen, Sidhanth Holalkere, Anna Asch, Aaron Lou, Ser-Nam Lim, and Christopher De Sa. Riemannian residual neural networks. In Advances in Neural Information Processing Systems, volume 36. 2023.
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Or Yair, Mirela Ben-Chen, and Ronen Talmon. Parallel transport on the cone manifold of spd matrices for domain adaptation. IEEE Transactions on Signal Processing, 67(7):1797–1811, 2019.
Yann Thanwerdas and Xavier Pennec. O(n)-invariant riemannian metrics on spd matrices. Linear Algebra and its Applications, 661:163–201, 2023. URL: https://www.sciencedirect.com/science/article/pii/S0024379522004360, doi:https://doi.org/10.1016/j.laa.2022.12.009.
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Joseph Paillard, Joerg F. Hipp, and Denis A. Engemann. Green: a lightweight architecture using learnable wavelets and riemannian geometry for biomarker exploration. Patterns, 6(1):101091, 2025. doi:10.1016/j.patter.2024.101091.
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Denis A Engemann and Alexandre Gramfort. Automated model selection in covariance estimation and spatial whitening of meg and eeg signals. NeuroImage, 108:328–342, 2015. doi:10.1016/j.neuroimage.2014.12.040.
Bruno Aristimunha, Ce Ju, Antoine Collas, Florent Bouchard, Ammar Mian, Bertrand Thirion, Sylvain Chevallier, and Reinmar Kobler. Spdlearn: a geometric deep learning python library for neural decoding through trivialization. To be submitted, 2026. URL: spdlearn/spd_learn.
Mehdi Rahim, Bertrand Thirion, and Gaël Varoquaux. Population-shrinkage of covariance to estimate better brain functional connectivity. In Medical Image Computing and Computer-Assisted Intervention (MICCAI), 460–468. 2017. doi:10.1007/978-3-319-66182-7_53.
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Daniel Wilson, Robin Tibor Schirrmeister, Lukas AW Gemein, and Tonio Ball. Deep riemannian networks for eeg decoding. Imaging Neuroscience, 2025. doi:10.1162/imag_a_00511.
Yue-Ting Pan, Jing-Lun Chou, and Chun-Shu Wei. Matt: a manifold attention network for eeg decoding. In Advances in Neural Information Processing Systems, volume 35, 31116–31129. 2022. URL: https://proceedings.neurips.cc/paper_files/paper/2022/hash/c981fd12b1d5703f19bd8289da9fc996-Abstract-Conference.html.
Robin Tibor Schirrmeister, Jost Tobias Springenberg, Lukas Dominique Josef Fiederer, Martin Glasstetter, Katharina Eggensperger, Michael Tangermann, Frank Hutter, Wolfram Burgard, and Tonio Ball. Deep learning with convolutional neural networks for eeg decoding and visualization. Human Brain Mapping, 38(11):5391–5420, 2017. doi:10.1002/hbm.23730.
Michael Tangermann, Klaus-Robert Müller, Ad Aertsen, Niels Birbaumer, Christoph Braun, Clemens Brunner, Robert Leeb, Carsten Mehring, Kai J Miller, Gernot R Müller-Putz, and others. Review of the bci competition iv. Frontiers in Neuroscience, 6:55, 2012. doi:10.3389/fnins.2012.00055.
Jared A Nielsen, Brandon A Zielinski, P Thomas Fletcher, Andrew L Alexander, Nicholas Lange, Erin D Bigler, Janet E Lainhart, and Jeffrey S Anderson. Abnormal lateralization of functional connectivity between language and default mode regions in autism. Molecular Autism, 5:8, 2014. doi:10.1186/2040-2392-5-8.
Kamalaker Dadi, Mehdi Rahim, Alexandre Abraham, Darya Chyzhyk, Michael Milham, Bertrand Thirion, and Gaël Varoquaux. Benchmarking functional connectome-based predictive models for resting-state fmri. NeuroImage, 192:115–134, 2019. doi:10.1016/j.neuroimage.2019.02.062.
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Shanglin Li, Motoaki Kawanabe, and Reinmar J Kobler. SPDIM: source-free unsupervised conditional and label shift adaptation in EEG. In The Thirteenth International Conference on Learning Representations. 2025. URL: https://openreview.net/forum?id=CoQw1dXtGb.
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Mehrtash Harandi, Mathieu Salzmann, and Richard Hartley. Dimensionality reduction on spd manifolds: the emergence of geometry-aware methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 40(1):48–62, 2017. doi:10.1109/TPAMI.2017.2655048.
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BibTeX Entries#
For convenience, here are BibTeX entries for citing the main works:
@inproceedings{huang2017riemannian,
title={A Riemannian network for SPD matrix learning},
author={Huang, Zhiwu and Van Gool, Luc},
booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
volume={31},
number={1},
year={2017}
}
@inproceedings{brooks2019riemannian,
title={Riemannian batch normalization for SPD neural networks},
author={Brooks, Daniel and Schwander, Olivier and Barbaresco, Fr{\'e}d{\'e}ric and
Schneider, Jean-Yves and Cord, Matthieu},
booktitle={Advances in Neural Information Processing Systems},
volume={32},
year={2019}
}
@inproceedings{kobler2022spd,
title={SPD domain-specific batch normalization to crack interpretable
unsupervised domain adaptation in EEG},
author={Kobler, Reinmar J and Hirayama, Jun-ichiro and Zhao, Qibin and Kawanabe, Motoaki},
booktitle={Advances in Neural Information Processing Systems},
volume={35},
pages={6219--6235},
year={2022}
}
@inproceedings{pan2022matt,
title={MAtt: A manifold attention network for EEG decoding},
author={Pan, Yue-Ting and Chou, Jing-Lun and Wei, Chun-Shu},
booktitle={Advances in Neural Information Processing Systems},
volume={35},
pages={31116--31129},
year={2022}
}
@article{paillard2024green,
title={GREEN: a lightweight architecture using learnable wavelets and
Riemannian geometry for biomarker exploration},
author={Paillard, Joseph and Hipp, Joerg F. and Engemann, Denis A.},
journal={Patterns},
volume={6},
number={1},
pages={101153},
year={2025},
doi={10.1016/j.patter.2025.101153}
}
@article{ju2022tensor,
title={Tensor-CSPNet: A Novel Geometric Deep Learning Framework for
Motor Imagery Classification},
author={Ju, Ce and Guan, Cuntai},
journal={IEEE Transactions on Neural Networks and Learning Systems},
volume={34},
number={12},
pages={10955--10969},
year={2023},
doi={10.1109/TNNLS.2022.3172108}
}
@article{carrara2024eegspd,
title={Geometric neural network based on phase space for BCI-EEG decoding},
author={Carrara*, Igor and Aristimunha*, Bruno and Corsi, Marie-Constance and
de Camargo, Raphael Y. and Chevallier, Sylvain and Papadopoulo, Th{\'e}odore},
journal={Journal of Neural Engineering},
volume={21},
number={6},
pages={016049},
year={2024},
doi={10.1088/1741-2552/ad88a2}
}
@article{ju2026spdmatrixlearningneuroimaging,
title={SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges},
author={Ce Ju and Reinmar Kobler and Antoine Collas and Motoaki Kawanabe and Cuntai Guan and Bertrand Thirion},
year={2026},
journal={arXiv preprint arXiv2504.18882},
url={https://arxiv.org/abs/2504.18882},
}
Citing SPD Learn#
If you use SPD Learn in your research, please cite:
@article{aristimunha2025spdlearn,
title={SPDlearn: A Geometric Deep Learning Python Library for
Neural Decoding Through Trivialization},
author={Aristimunha, Bruno and Ju, Ce and Collas, Antoine and
Bouchard, Florent and Thirion, Bertrand and
Chevallier, Sylvain and Kobler, Reinmar},
journal={To be submitted},
year={2026},
url={https://github.com/spdlearn/spd_learn}
}
See also
Theory – Theoretical foundations
API Reference – API reference for implemented models
Applied Examples – Practical examples