Skip to main content
SpringerLink
Log in

Using Geometry of the Set of Symmetric Positive Semidefinite Matrices to Classify Structural Brain Networks

  • Conference paper
  • First Online:
Computational Aspects and Applications in Large-Scale Networks (NET 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 247))

Included in the following conference series:

  • 506 Accesses

Abstract

This paper presents a method of symmetric positive semidefinite (SPSD) matrices classification and its application to the analysis of structural brain networks (connectomes). Structural connectomes are modeled as weighted graphs in which edge weights are proportional to the number of streamline connections between brain regions detected by a tractography algorithm. The construction of structural brain networks does not typically guarantee that their adjacency matrices lie in some topological space with known properties. This makes them differ from functional connectomes—correlation matrices representing co-activation of brain regions, which are usually symmetric positive definite (SPD). Here, we propose to transform structural connectomes by taking their normalized Laplacians prior to any analysis, to put them into a space of symmetric positive semidefinite (SPSD) matrices, and apply methods developed for manifold-valued data. The geometry of the SPD matrix manifold is well known and used in many classification algorithms. Here, we expand existing SPD matrix-based algorithms to the SPSD geometry and develop classification pipelines on SPSD normalized Laplacians of structural connectomes. We demonstrate the performance of the proposed pipeline on structural brain networks reconstructed from the Alzheimer‘s Disease Neuroimaging Initiative (ADNI) data.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barachant, A., Bonnet, S., Congedo, M., Jutten, C.: Multiclass brain-computer interface classification by Riemannian geometry. IEEE Trans. Biomed. Eng. 59(4), 920–928 (2012)

    Article  Google Scholar 

  2. Barachant, A., Bonnet, S., Congedo, M., Jutten, C.: Classification of covariance matrices using a Riemannian-based kernel for BCI applications. Neurocomputing 112, 172–178 (2013)

    Article  Google Scholar 

  3. Bonnabel, S., Sepulchre, R.: Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31(3), 1055–1070 (2009)

    Article  MathSciNet  Google Scholar 

  4. Brown, C.J., Hamarneh, G.: Machine learning on human connectome data from MRI. arXiv:1611.08699 (2016)

  5. Desikan, R.S., Ségonne, F., Fischl, B., Quinn, B.T., Dickerson, B.C., Blacker, D., Buckner, R.L., et al.: An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. Neuroimage 31(3), 968–980 (2006)

    Article  Google Scholar 

  6. Dodero, L., Minh, H.Q., San Biagio, M., Murino, V., Sona, D.: Kernel-based classification for brain connectivity graphs on the Riemannian manifold of positive definite matrices. In: 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI), pp. 42–45. IEEE (2015)

    Google Scholar 

  7. Faraut, J., Korányi, A.: Analysis on symmetric cones (1994)

    Google Scholar 

  8. Fischl, B.: Freesurfer. Neuroimage 62(2), 774–781 (2012)

    Article  Google Scholar 

  9. Garyfallidis, E., Brett, M., Amirbekian, B., Rokem, A., Van Der Walt, S., Descoteaux, M., Nimmo-Smith, I.: Dipy, a library for the analysis of diffusion MRI data. Front. Neuroinform. 8, 8 (2014)

    Article  Google Scholar 

  10. Jayasumana, S., Hartley, R., Salzmann, M., Li, H., Harandi, M.: Kernel methods on the riemannian manifold of symmetric positive definite matrices. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 73–80 (2013)

    Google Scholar 

  11. Krivov, E., Belyaev, M.: Dimensionality reduction with isomap algorithm for EEG covariance matrices. In: 2016 4th International Winter Conference on Brain-Computer Interfaces (BCI), pp. 1–4. IEEE (2016)

    Google Scholar 

  12. Kruskal, J.B., Wish, M.: Multidimensional Scaling, vol. 11. Sage (1978)

    Google Scholar 

  13. Kurmukov, A., Dodonova, Y., Zhukov, L.: Classification of normal and pathological brain networks based on similarity in graph partitions. In: 2016 IEEE 16th International Conference on Data Mining Workshops (ICDMW), pp. 107–112. IEEE (2016)

    Google Scholar 

  14. Ng, B., Varoquaux, G., Poline, J.B., Greicius, M., Thirion, B.: Transport on Riemannian manifold for connectivity-based brain decoding. IEEE Trans. Med. Imaging 35(1), 208–216 (2016)

    Article  Google Scholar 

  15. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006)

    Article  Google Scholar 

  16. Petrov, D., Dodonova, Y., Zhukov, L., Belyaev, M.: Boosting connectome classification via combination of geometric and topological normalizations. In: 2016 International Workshop on Pattern Recognition in Neuroimaging (PRNI), pp. 1–4. IEEE (2016)

    Google Scholar 

  17. Scholkopf, B., Smola, A.J.: Learning with kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press (2001)

    Google Scholar 

  18. Slavakis, K., Salsabilian, S., Wack, D.S., Muldoon, S.F.: Clustering time-varying connectivity networks by Riemannian geometry: the brain-network case. In: 2016 IEEE Statistical Signal Processing Workshop (SSP), pp. 1–5. IEEE (2016)

    Google Scholar 

  19. Smith, S.M., Vidaurre, D., Beckmann, C.F., Glasser, M.F., Jenkinson, M., et al.: Functional connectomics from resting-state fMRI. Trends Cognitive Sci. 17(12), 666–682 (2013)

    Article  Google Scholar 

  20. Tax, C.M., Jeurissen, B., Vos, S.B., Viergever, M.A., Leemans, A.: Recursive calibration of the fiber response function for spherical deconvolution of diffusion MRI data. Neuroimage 86, 67–80 (2014)

    Article  Google Scholar 

  21. Tenenbaum, J.B., De Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)

    Article  Google Scholar 

Download references

Acknowledgements

Some data used in preparing this article were obtained from the Alzheimers Disease Neuroimaging Initiative (ADNI) database. A complete listing of ADNI investigators and imaging protocols may be found at http://www.adni.loni.usc.edu.The results of Sects. 2–6 are based on the scientific research conducted at IITP RAS and supported by the Russian Science Foundation under grant 17-11-01390.

Author information

Authors and Affiliations

  1. Skolkovo Institute of Science and Technology, Moscow, Russia

    Mikhail Belyaev, Daria Belyaeva & Egor Krivov

  2. Kharkevich Institute for Information Transmission Problems, Moscow, Russia

    Mikhail Belyaev, Yulia Dodonova, Daria Belyaeva, Egor Krivov & Boris Gutman

  3. Moscow Institute of Physics and Technology, Moscow, Russia

    Egor Krivov

  4. Imaging Genetics Center, Stevens Neuroimaging and Informatics Institute, University of Southern California, Marina del Rey, CA, USA

    Boris Gutman, Joshua Faskowitz, Neda Jahanshad & Paul Thompson

Authors
  1. Mikhail Belyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yulia Dodonova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daria Belyaeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Egor Krivov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Boris Gutman
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Joshua Faskowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Neda Jahanshad
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Paul Thompson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail Belyaev .

Editor information

Editors and Affiliations

  1. Higher School of Economics, National Research University, Nizhny Novgorod, Russia

    Valery A. Kalyagin

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  3. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Oleg Prokopyev

  4. Higher School of Economics, National Research University, Nizhny Novgorod, Russia

    Irina Utkina

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Belyaev, M. et al. (2018). Using Geometry of the Set of Symmetric Positive Semidefinite Matrices to Classify Structural Brain Networks. In: Kalyagin, V., Pardalos, P., Prokopyev, O., Utkina, I. (eds) Computational Aspects and Applications in Large-Scale Networks. NET 2016. Springer Proceedings in Mathematics & Statistics, vol 247. Springer, Cham. https://doi.org/10.1007/978-3-319-96247-4_18

Download citation

Publish with us

Policies and ethics

Search

Navigation