Skip to main content

Advertisement

SpringerLink
Log in

Review of Riemannian Distances and Divergences, Applied to SSVEP-based BCI

  • Original Article
  • Published:
Neuroinformatics Aims and scope Submit manuscript

Abstract

The firstgeneration of brain-computer interfaces (BCI) classifies multi-channel electroencephalographic (EEG) signals, enhanced by optimized spatial filters.The second generation directly classifies covariance matrices estimated on EEG signals, based on straightforward algorithms such as the minimum-distance-to-Riemannian-mean (MDRM). Classification results vary greatly depending on the chosen Riemannian distance or divergence, whose definitions and reference implementations are spread across a wide mathematical literature. This paper reviews all the Riemannian distances and divergences to process covariance matrices, with an implementation compatible with BCI constraints. The impact of using different metrics is assessed on a steady-state visually evoked potentials (SSVEP) dataset, evaluating centers of classes and classification accuracy. Riemannian approaches embed crucial properties to process EEG data. The Riemannian centers of classes outperform Euclidean ones both in offline and online setups. Some Riemannian distances and divergences have better performances in terms of classification accuracy, while others have appealing computational efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Afsari, B. (2011). Riemannian Lp center of mass: existence, uniqueness, and convexity. Proc. Amer. Math. Soc., 139(2), 655–673.

    Google Scholar 

  • Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904–924.

    Google Scholar 

  • Álvarez-Esteban, P., del Barrio, E., Cuesta-Albertos, J., & Matrán, C. (2016). A fixed-point approach to barycenters in Wasserstein space. Journal of Mathematical Analysis and Applications, 441, 744–762.

    Google Scholar 

  • Ando, T., Li, C.K., & Mathias, R. (2004). Geometric means. Linear Algebra and its Applications, 385, 305–334.

    Google Scholar 

  • Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2007). Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl., 29(1), 328–347.

    Google Scholar 

  • Barachant, A., Bonnet, S., Congedo, M., & Jutten, C. (2012). Multiclass brain–computer interface classification by Riemannian geometry. IEEE Transactions on Biomedical Engineering, 59(4), 920–928.

    PubMed  Google Scholar 

  • Barachant, A., Andreev, A., & Congedo, M. (2013). The Riemannian potato: An automatic and adaptive artifact detection method for online experiments using Riemannian geometry. In Proceedings of TOBI Workshop, (Vol. IV pp. 19–20).

  • Barachant, A., Jayaram, V., Chevallier, S., & Rodrigues, P. (2019). MOABB framework. https://github.com/NeuroTechX/moabb.

  • Bertrand-Lalo, R. (2020). Online SSVEP classification in Python with TimeFlux. https://github.com/bertrandlalo/timeflux_ssvep.

  • Bhatia, R. (2009). Positive definite matrices, vol 16. Princeton University Press.

  • Bhattacharyya, A. (1943). On a measure of divergence between two statistical populations defined by their probability distribution. Bull. Calcutta Math. Soc., 35, 99–109.

    Google Scholar 

  • Cartan, E. (1929). Groupes simples clos et ouverts et géométrie riemannienne. Journal de mathé,matiques pures et appliquées, pp. 1–34.

  • Chebbi, Z., & Moakher, M. (2012). Means of Hermitian positive-definite matrices based on the log-determinant α-divergence function. Linear Algebra and its Applications, 436(7), 1872–1889.

    Google Scholar 

  • Chen, X., Chen, Z., Gao, S., & Gao, X. (2014). A high-ITR SSVEP-based BCI speller. Brain-Computer Interfaces, 1(3-4), 181–191.

    Google Scholar 

  • Chen, X., Wang, Y., Gao, S., Jung, T., & Gao, S. (2015a). Filter bank canonical correlation analysis for implementing a high-speed SSVEP-based brain-computer interface. J Neural Eng.

  • Chen, X., Wang, Y., Nakanishi, M., Gao, X., Jung, T., & Gao, S. (2015b). High-speed spelling with a noninvasive brain–computer interface. Proceedings of the National Academy of Sciences, 112(44), E6058–E6067.

    CAS  Google Scholar 

  • Cherian, A., Sra, S., Banerjee, A., & Papanikolopoulos, N. (2011). Efficient similarity search for covariance matrices via the Jensen-Bregman LogDet divergence. In: Int Conf Computer Vision, IEEE, pp. 2399–2406.

  • Chevallier, S. (2017). SSVEP data. https://github.com/sylvchev/dataset-ssvep-exoskeleton.

  • Chevallier, S. (2020). Offline SSVEP classification in Python. https://github.com/alexandrebarachant/pyRiemann/tree/master/examples/SSVEP.

  • Chevallier, S., Kalunga, E., Barthélemy, Q., & Yger, F. (2018). Brain computer interfaces handbook: Technological and theoretical advances, CRC Press, chap 19 - Riemannian classification for SSVEP based BCI: Offline versus online implementations, pp. 371–396.

  • Cichocki, A., & Amari, S. (2010). Families of alpha-beta-and gamma-divergences: Flexible and robust measures of similarities. Entropy, 12(6), 1532–1568.

    Google Scholar 

  • Congedo, M., Barachant A, & Bhatia R. (2017a). Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review. Brain-Computer Interfaces, 4, 1–20.

    Google Scholar 

  • Congedo, M., Barachant, A., & Koopaei, E. (2017b). Fixed point algorithms for estimating power means of positive definite matrices. IEEE Trans. Signal Process., 65, 2211–2220.

    Google Scholar 

  • Dhillon, I.S., & Tropp, J.A. (2007). Matrix nearness problems with Bregman divergences. SIAM J Matrix Anal Appl, 29(4), 1120–1146.

    Google Scholar 

  • Fletcher, P., & Joshi, S. (2004). Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Computer vision and mathematical methods in medical and biomedical image analysis, LNCS, vol. 3117, Springer, pp. 87–98.

  • Gergondet, P., & Kheddar, A. (2015). SSVEP stimuli design for object-centric BCI. Brain-Computer Interfaces, 2(1), 11–28.

    Google Scholar 

  • Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning. Springer series in statistics. New York: Springer.

    Google Scholar 

  • Herrmann, C.S. (2001). Human EEG responses to 1-100 Hz flicker: Resonance phenomena in visual cortex and their potential correlation to cognitive phenomena. Experimental Brain Research, 137(3-4), 346–353.

    CAS  PubMed  Google Scholar 

  • Hosni, S.M., Shedeed, H.A., Mabrouk, M.S., & Tolba, M.F. (2018). EEG-EOG based virtual keyboard: Toward hybrid brain computer interface. Neuroinformatics.

  • Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 186, 453–461.

    CAS  Google Scholar 

  • Johannes, M.G., Pfurtscheller, G., & Flyvbjerg, H. (1999). Designing optimal spatial filters for single-trial EEG classification in a movement task. Clinical Neurophysiology, 110(5), 787–798.

    Google Scholar 

  • Kalunga, E., Djouani, K., Hamam, Y., Chevallier, S., & Monacelli, E. (2013). SSVEP enhancement based on Canonical Correlation Analysis to improve BCI performances. In AfriCon, (Vol. 2013 pp. 1–5).

  • Kalunga, E., Chevallier, S., Barthélemy, Q., Djouani, K., Hamam, Y., & Monacelli, E. (2015). From Euclidean to Riemannian Means: Information Geometry for SSVEP Classification. In: Geometric Science of Information, 9389, Springer International Publishing, pp. 595–604.

  • Kalunga, E.K. (2015). Online SSVEP classification in Matlab. https://github.com/emmanuelkalunga/Online-SSVEP.

  • Kalunga, E.K. (2018). Offline SSVEP classification in Matlab. https://github.com/emmanuelkalunga/Offline-Riemannian-SSVEP.

  • Kalunga, E.K., Chevallier, S., Rabreau, O., & Monacelli, E. (2014). Hybrid interface: Integrating BCI in multimodal human-machine interfaces. In: 2014 IEEE/ASME int conf advanced intelligent mechatronics (AIM), pp. 530–535.

  • Kalunga, E.K., Chevallier, S., Barthélemy, Q., Djouani, K., Monacelli, E., & Hamam, Y. (2016). Online SSVEP-based BCI using Riemannian geometry. Neurocomputing, 191, 55–68.

    Google Scholar 

  • Lim, Y., & Pálfia, M. (2012). Matrix power means and the Karcher mean. Journal of Functional Analysis, 262(4), 1498–1514.

    Google Scholar 

  • Lin, Z., Zhang, C., Wu, W., & Gao, X. (2006). Frequency recognition based on canonical correlation analysis for SSVEP-based BCIs. IEEE Transactions on Biomedical Engineering, 53(12), 2610–2614.

    PubMed  Google Scholar 

  • Lin, Z., Zhang, C., Wu, W., & Gao, X. (2007). Frequency recognition based on canonical correlation analysis for SSVEP-based BCIs. IEEE Transactions on Biomedical Engineering, 53(12), 2610–2614.

    Google Scholar 

  • Lotte, L., Bougrain, L., Cichocki, A., Clerc, M., Congedo, M., Rakotomamonjy, A., & Yger, F. (2018). A review of classification algorithms for EEG-based brain-computer interfaces: A 10 year update. J. Neural Eng., 15(031), 005.

    Google Scholar 

  • Martin, H., Chevallier, S., & Monacelli, E. (2012). Fast calibration of hand movement-based interface for arm exoskeleton control. In: European symposium on artificial neural networks (ESANN), pp. 573–578.

  • Massart, E.M., & Chevallier, S. (2017). Inductive means and sequences applied to online classification of eeg. In: International conference on geometric science of information, Springer, pp 763–770.

  • McFarland, D., Sarnacki, W., & Wolpaw, J. (2003). Brain–computer interface (bci) operation: Optimizing information transfer rates. Biological psychology, 63(3), 237–251.

    PubMed  Google Scholar 

  • McFarland, D., Daly, J., Boulay, C., & Parvaz, M. (2017). Therapeutic applications of BCI technologies. Brain-Computer Interfaces, 4, 1–2.

    Google Scholar 

  • Meinel, A., Castaṅo-Candamil, S., Blankertz, B., Lotte, F., & Tangermann, M. (2019). Characterizing regularization techniques for spatial filter optimization in oscillatory EEG regression problems. Neuroinformatics, 17 (2), 235–251.

    PubMed  Google Scholar 

  • Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl., 26(3), 735–747.

    Google Scholar 

  • Moakher, M., & Batchelor, P.G. (2006). 17. Symmetric positive-definite matrices: From geometry to applications and visualization, (pp. 285–298). Berlin: Springer.

  • Nakanishi, M., Wang, Y., Wang, Y., Mitsukura, Y., & Jung, T. (1450). High-speed brain speller using steady-state visual evoked potentials. International Journal of Neural Systems, 24(06), 019.

    Google Scholar 

  • Nam, C.S., Choi, I., Wadeson, A., & Whang, M. (2018). Brain-computer interfaces handbook: Technological and theoretical advances, CRC Press, chap Brain-Computer Interface An Emerging Interaction Technology.

  • Niedermeyer, E., & Silva, F.H.L.D. (2005). Electroencephalography: Basic Principles, Clinical Applications, and Related Fields. Lippincott Williams & Wilkins.

  • Nielsen, F., & Bhatia, R. (2012). Matrix information geometry. Berlin: Springer Publishing Company Incorporated.

    Google Scholar 

  • Nielsen, F., & Nock, R. (2009). Sided and symmetrized Bregman centroids. IEEE Transactions on Information Theory, 55(6), 2882–2904.

    Google Scholar 

  • Nielsen, F., Nock, R., & Amari, S. (2014). On clustering histograms with k-means by using mixed α-divergences. Entropy, 16(6), 3273–3301.

    Google Scholar 

  • Pennec, X., Fillard, P., & Ayache, N. (2006). A Riemannian Framework for Tensor Computing. International Journal of Computer Vision, 66(1), 41–66.

    Google Scholar 

  • Quitadamo, L.R., Marciani, M.G., Cardarilli, G.C., & Bianchi, L. (2008). Describing different brain computer interface systems through a unique model: A UML implementation. Neuroinformatics, 6(2), 81–96.

    PubMed  Google Scholar 

  • Rivet, B., Souloumiac, A., Attina, V., & Gibert, G. (2009). xDAWN algorithm to enhance evoked potentials: Application to brain-computer interface. IEEE Trans. Biomed. Eng.., 56(8), 2035–2043.

    PubMed  Google Scholar 

  • Sra, S. (2016). Positive definite matrices and the S-divergence. Proceedings of the American Mathematical Society, 144(7), 2787–2797.

    Google Scholar 

  • Tomioka, R., Aihara, K., & Müller, K.R. (2007). Logistic regression for single trial EEG classification. In Advances in neural information processing systems (NIPS), (Vol. 19 pp. 1377–1384).

  • Verschore, H., Kindermans, P.J., Verstraeten, D., & Schrauwen, B. (2012). Dynamic stopping improves the speed and accuracy of a P300 speller. In: Artificial neural networks and machine learning–ICANN, Springer, pp. 661–668.

  • Vialatte, F.B., Maurice, M., Dauwels, J., & Cichocki, A. (2010). Steady-state visually evoked potentials: Focus on essential paradigms and future perspectives. Progress in Neurobiology, 90(4), 418–438.

    PubMed  Google Scholar 

  • Villani, C. (2008). Optimal transport: old and new, vol 338. Springer Science & Business Media.

  • Volosyak, I., Valbuena, D., Luth, T., Malechka, T., & Graser, A. (2011). BCI demographics II: How many (and what kinds of) people can use a high-frequency SSVEP BCI? IEEE Trans. Neural Syst. Rehabil. Eng., 19 (3), 232–239.

    PubMed  Google Scholar 

  • Wang, S., & James, C.J. (2006). Enhancing Evoked Responses for BCI Through Advanced ICA Techniques. In: Advances in medical, signal and information processing (MEDSIP), pp. 1–4.

  • Wang, Y., Zhang, Z., Gao, X., & Gao, S. (2004). Lead selection for SSVEP- based brain-computer interface. In Int Conf IEEE engineering in medicine and biology society, (Vol. 2 pp. 4507–4510).

  • Yang, B., Zhang, T., Zhang, Y., Liu, W., Wang, J., & Duan, K. (2017). Removal of electrooculogram artifacts from electroencephalogram using canonical correlation analysis with ensemble empirical mode decomposition. Cognitive Computation, 9(5), 626–633.

    Google Scholar 

  • Yang, Y., Bloch, I., Chevallier, S., & Wiart, J. (2016). Subject-specific channel selection using time information for motor imagery brain–computer interfaces. Cognitive Computation, 8(3), 505–518.

    Google Scholar 

  • Yger, F., Berar, M., & Lotte, F. (2016). Riemannian approaches in brain-computer interfaces: A review. IEEE Trans. Neural Syst. Rehabil. Eng., 25(10), 1753–1762.

    PubMed  Google Scholar 

  • Zhang, Y., Zhou, G., Jin, J., Wang, X., & Cichocki, A. (2015). Ssvep recognition using common feature analysis in brain–computer interface. Journal of Neuroscience Methods, 244, 8–15.

    PubMed  Google Scholar 

  • Zhu, D., Bieger, J., Molina, G.G., & Aarts, R.M. (2010). A survey of stimulation methods used in SSVEP-based BCIs. Intell. Neuroscience, 2010, 1–12.

    Google Scholar 

Download references

Acknowledgments

This work has been done without any support of the ANR or the ERC.

Author information

Authors and Affiliations

  1. Université Paris-Saclay, UVSQ, LISV, 78124, Vélizy-Villacoublay, France

    S. Chevallier & E. Monacelli

  2. Orderin, Cape Town, South Africa

    E. K. Kalunga

  3. Foxstream, 69120, Vaulx-en-Velin, France

    Q. Barthélemy

Authors
  1. S. Chevallier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. K. Kalunga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Q. Barthélemy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Monacelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Chevallier.

Ethics declarations

Conflict of interests

At the time this study was carried out, Emmanuel Kalunga was an employee of VASTech company, Stellenbosch, South Africa; and Quentin Barthélemy was an employee of Mensia Technologies S.A., Paris, France. Sylvain Chevallier and Eric Monacelli declare that they have no conflict of interest. This study was conducted by the authors without any specific funding or grant.

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

(PDF 234 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chevallier, S., Kalunga, E.K., Barthélemy, Q. et al. Review of Riemannian Distances and Divergences, Applied to SSVEP-based BCI. Neuroinform 19, 93–106 (2021). https://doi.org/10.1007/s12021-020-09473-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12021-020-09473-9

Keywords

Search

Navigation