Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment

  • Artjom ZernEmail author
  • Matthias Zisler
  • Freddie Åström
  • Stefania Petra
  • Christoph Schnörr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11269)


Manifold models of image features abound in computer vision. We present a novel approach that combines unsupervised computation of representative manifold-valued features, called labels, and the spatially regularized assignment of these labels to given manifold-valued data. Both processes evolve dynamically through two Riemannian gradient flows that are coupled. The representation of labels and assignment variables are kept separate, to enable the flexible application to various manifold data models. As a case study, we apply our approach to the unsupervised learning of covariance descriptors on the positive definite matrix manifold, through spatially regularized geometric assignment.



This work was supported by the German Research Foundation (DFG), grant GRK 1653.


  1. 1.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29(1), 328–347 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Åström, F., Petra, S., Schmitzer, B., Schnörr, C.: Image labeling by assignment. J. Math. Imaging Vis. 58(2), 211–238 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  4. 4.
    Chebbi, Z., Moakher, M.: Means of Hermitian positive-definite matrices based on the log-determinant \(\alpha \)-divergence function. Linear Algebra Appl. 436(7), 1872–1889 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cherian, A., Sra, S.: Positive definite matrices: data representation and applications to computer vision. In: Minh, H.Q., Murino, V. (eds.) Algorithmic Advances in Riemannian Geometry and Applications. ACVPR, pp. 93–114. Springer, Cham (2016). Scholar
  6. 6.
    Cherian, A., Sra, S., Banerjee, A., Papanikolopoulos, N.: Jensen-Bregman LogDet divergence with application to efficient similarity search for covariance matrices. IEEE PAMI 35(9), 2161–2174 (2013)CrossRefGoogle Scholar
  7. 7.
    Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Trans. Patt. Anal. Mach. Intell. 24(5), 603–619 (2002)CrossRefGoogle Scholar
  8. 8.
    Fukunaga, K., Hostetler, L.: The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans. Inform. Theory 21(1), 32–40 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Har-Peled, S.: Geometric Approximation Algorithms. AMS, Providence (2011)CrossRefGoogle Scholar
  10. 10.
    Harandi, M., Hartley, R., Lovell, B., Sanderson, C.: Sparse coding on symmetric positive definite manifolds using Bregman divergences. IEEE Trans. Neural Netw. Learn. Syst. 27(6), 1294–1306 (2016)CrossRefGoogle Scholar
  11. 11.
    Hofmann, T., Schölkopf, B., Smola, A.J.: Kernel methods in machine learning. Ann. Stat. 36(3), 1171–1220 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hühnerbein, R., Savarino, F., Åström, F., Schnörr, C.: Image labeling based on graphical models using Wasserstein messages and geometric assignment. SIAM J. Imaging Sci. 11(2), 1317–1362 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Heidelberg (2009). Scholar
  15. 15.
    Savarino, F., Hühnerbein, R., Åström, F., Recknagel, J., Schnörr, C.: Numerical integration of Riemannian gradient flows for image labeling. In: Lauze, F., Dong, Y., Dahl, A.B. (eds.) SSVM 2017. LNCS, vol. 10302, pp. 361–372. Springer, Cham (2017). Scholar
  16. 16.
    Sra, S.: Positive Definite Matrices and the Symmetric Stein Divergence. CoRR abs/1110.1773 (2013)Google Scholar
  17. 17.
    Subbarao, R., Meer, P.: Nonlinear mean shift over Riemannian manifolds. Int. J. Comput. Vis. 84(1), 1–20 (2009)CrossRefGoogle Scholar
  18. 18.
    Teboulle, M.: A unified continuous optimization framework for center-based clustering methods. J. Mach. Learn. Res. 8, 65–102 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Turaga, P., Srivastava, A. (eds.): Riemannian Computing in Computer Vision. Springer, Cham (2016). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Artjom Zern
    • 1
    Email author
  • Matthias Zisler
    • 1
  • Freddie Åström
    • 1
  • Stefania Petra
    • 2
  • Christoph Schnörr
    • 1
  1. 1.Image and Pattern Analysis GroupHeidelberg UniversityHeidelbergGermany
  2. 2.Mathematical Imaging GroupHeidelberg UniversityHeidelbergGermany

Personalised recommendations