Estimating Latent Brain Sources with Low-Rank Representation and Graph Regularization

  • Feng Liu
  • Shouyi WangEmail author
  • Jing Qin
  • Yifei Lou
  • Jay Rosenberger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11309)


To infer latent brain source activation patterns under different cognitive tasks is an integral step to understand how our brain works. Traditional electroencephalogram (EEG) Source Imaging (ESI) methods usually do not distinguish task-related and spurious non-task-related sources that jointly generate EEG signals, which inevitably yield misleading reconstructed activation patterns. In this research, we assume that the task-related source signal intrinsically has a low-rank property, which is exploited to infer the true task-related EEG sources location. Although the true task-related source signal is sparse and low-rank, the contribution of spurious sources scattering over the source space with intermittent activation patterns makes the actual source space lose the low-rank property. To reconstruct a low-rank true source, we propose a novel ESI model that involves a spatial low-rank representation and a temporal Laplacian graph regularization, the latter of which guarantees the temporal smoothness of the source signal and eliminate the spurious ones. To solve the proposed model, an augmented Lagrangian objective function is formulated and an algorithm in the framework of alternating direction method of multipliers (ADMM) is proposed. Numerical results illustrate the effectivenesks of the proposed method in terms of reconstruction accuracy with high efficiency.


EEG Source Imaging Low rank representation Graph Regularization Alternating direction method of multiplier (ADMM) 



This work has been partially supported by the NSF funding under grant number CMMI-1537504 and DMS-1522786. The research of Jing Qin is supported by the NSF grant DMS-1818374.


  1. 1.
    Liu, F., Xiang, W., Wang, S., Lega, B.: Prediction of seizure spread network via sparse representations of overcomplete dictionaries. In: Ascoli, G.A., Hawrylycz, M., Ali, H., Khazanchi, D., Shi, Y. (eds.) BIH 2016. LNCS (LNAI), vol. 9919, pp. 262–273. Springer, Cham (2016). Scholar
  2. 2.
    Ding, L.: Reconstructing cortical current density by exploring sparseness in the transform domain. Phys. Med. Biol. 54(9), 2683 (2009)CrossRefGoogle Scholar
  3. 3.
    Grech, R., Cassar, T., Muscat, J., Camilleri, K.P., Fabri, S.G., Zervakis, M., Xanthopoulos, P., Sakkalis, V., Vanrumste, B.: Review on solving the inverse problem in EEG source analysis. J. Neuroeng. Rehabil. 5(1), 1 (2008)CrossRefGoogle Scholar
  4. 4.
    He, B., Sohrabpour, A., Brown, E., Liu, Z.: Electrophysiological source imaging: a noninvasive window to brain dynamics. Annu. Rev. Biomed. Eng. 20, 171–196 (2018)CrossRefGoogle Scholar
  5. 5.
    Gramfort, A., Kowalski, M., Hämäläinen, M.: Mixed-norm estimates for the M/EEG inverse problem using accelerated gradient methods. Phys. Med. Biol. 57(7), 1937 (2012)CrossRefGoogle Scholar
  6. 6.
    Gramfort, A., Strohmeier, D., Haueisen, J., Hämäläinen, M.S., Kowalski, M.: Time-frequency mixed-norm estimates: Sparse M/EEG imaging with non-stationary source activations. NeuroImage 70, 410–422 (2013)CrossRefGoogle Scholar
  7. 7.
    Liu, F., Rosenberger, J., Lou, Y., Hosseini, R., Su, J., Wang, S.: Graph regularized EEG source imaging with in-class consistency and out-class discrimination. IEEE Trans. Big Data 3(4), 378–391 (2017)CrossRefGoogle Scholar
  8. 8.
    Qin, J., Liu, F., Wang, S., Rosenberger, J.: EEG source imaging based on spatial and temporal graph structures. In: International Conference on Image Processing Theory, Tools and Applications (2017)Google Scholar
  9. 9.
    Liu, F., Hosseini, R., Rosenberger, J., Wang, S., Su, J.: Supervised discriminative EEG brain source imaging with graph regularization. In: Descoteaux, M., Maier-Hein, L., Franz, A., Jannin, P., Collins, D.L., Duchesne, S. (eds.) MICCAI 2017. LNCS, vol. 10433, pp. 495–504. Springer, Cham (2017). Scholar
  10. 10.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefGoogle Scholar
  11. 11.
    Yin, M., Gao, J., Lin, Z.: Laplacian regularized low-rank representation and its applications. IEEE Trans. Pattern Anal. Mach. Intell. 38(3), 504–517 (2016)CrossRefGoogle Scholar
  12. 12.
    Cai, D., He, X., Han, J., Huang, T.S.: Graph regularized nonnegative matrix factorization for data representation. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1548–1560 (2011)CrossRefGoogle Scholar
  13. 13.
    Michel, V., Gramfort, A., Varoquaux, G., Eger, E., Thirion, B.: Total variation regularization for fMRI-based prediction of behavior. IEEE Trans. Med. Imaging 30(7), 1328–1340 (2011)CrossRefGoogle Scholar
  14. 14.
    Li, Y., Qin, J., Hsin, Y.L., Osher, S., Liu, W.: s-SMOOTH: sparsity and smoothness enhanced EEG brain tomography. Frontiers Neurosci. 10, 543 (2016)Google Scholar
  15. 15.
    Lin, Z., Liu, R., Su, Z.: Linearized alternating direction method with adaptive penalty for low-rank representation. In: Advances in Neural Information Processing Systems, pp. 612–620 (2011)Google Scholar
  16. 16.
    Cai, J.F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nie, F., Huang, H., Cai, X., Ding, C.H.: Efficient and robust feature selection via joint \(\ell _{2,1}\)-norms minimization. In: Advances in Neural Information Processing Systems, pp. 1813–1821 (2010)Google Scholar
  18. 18.
    Du, S., Ma, Y., Ma, Y.: Graph regularized compact low rank representation for subspace clustering. Knowl.-Based Syst. 118, 56–69 (2017)CrossRefGoogle Scholar
  19. 19.
    Yin, M., Gao, J., Lin, Z., Shi, Q., Guo, Y.: Dual graph regularized latent low-rank representation for subspace clustering. IEEE Trans. Image Process. 24(12), 4918–4933 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hämäläinen, M.S., Ilmoniemi, R.J.: Interpreting magnetic fields of the brain: minimum norm estimates. Med. Biol. Eng. Comput. 32(1), 35–42 (1994)CrossRefGoogle Scholar
  21. 21.
    Pascual-Marqui, R.D., et al.: Standardized low-resolution brain electromagnetic tomography (sloreta): technical details. Methods Find. Exp. Clin. Pharmacol. 24(Suppl D), 5–12 (2002)Google Scholar
  22. 22.
    Yang, A.Y., Sastry, S.S., Ganesh, A., Ma, Y.: Fast \(\ell \) 1-minimization algorithms and an application in robust face recognition: a review. In: 2010 17th IEEE International Conference on Image Processing (ICIP), pp. 1849–1852. IEEE (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Feng Liu
    • 1
    • 2
  • Shouyi Wang
    • 3
    Email author
  • Jing Qin
    • 4
  • Yifei Lou
    • 5
  • Jay Rosenberger
    • 3
  1. 1.Massachusetts General HospitalHarvard Medical SchoolBostonUSA
  2. 2.Picower Institue of Learning and MemoryMITCambridgeUSA
  3. 3.Department of Industrial EngineeringUniversity of Texas at ArlingtonArlingtonUSA
  4. 4.Department of Mathematical SciencesMontana State UniversityBozemanUSA
  5. 5.Department of Mathematical SciencesUniversity of Texas at DallasRichardsonUSA

Personalised recommendations