Advertisement

Expanding the Family of Grassmannian Kernels: An Embedding Perspective

  • Mehrtash T. Harandi
  • Mathieu Salzmann
  • Sadeep Jayasumana
  • Richard Hartley
  • Hongdong Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8695)

Abstract

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g., support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show- is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing.

Keywords

Grassmann manifolds kernel methods Plücker embedding 

Supplementary material

978-3-319-10584-0_27_MOESM1_ESM.pdf (488 kb)
Electronic Supplementary Material (PDF 488 KB)

References

  1. 1.
    Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  2. 2.
    Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer (1984)Google Scholar
  3. 3.
    Chikuse, Y.: Statistics on Special Manifolds. Springer (2003)Google Scholar
  4. 4.
    Gopalan, R., Li, R., Chellappa, R.: Unsupervised adaptation across domain shifts by generating intermediate data representations. IEEE Trans. Pattern Analysis and Machine Intelligence (2014)Google Scholar
  5. 5.
    Hamm, J., Lee, D.D.: Grassmann discriminant analysis: a unifying view on subspace-based learning. In: Proc. Int. Conference on Machine Learning (ICML), pp. 376–383 (2008)Google Scholar
  6. 6.
    Harandi, M., Sanderson, C., Shen, C., Lovell, B.C.: Dictionary learning and sparse coding on grassmann manifolds: An extrinsic solution. In: Proc. Int. Conference on Computer Vision (ICCV) (December 2013)Google Scholar
  7. 7.
    Harandi, M.T., Sanderson, C., Shirazi, S., Lovell, B.C.: Graph embedding discriminant analysis on Grassmannian manifolds for improved image set matching. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2705–2712 (2011)Google Scholar
  8. 8.
    Helmke, U., Hüper, K., Trumpf, J.: Newtons’s method on Grassmann manifolds. Preprint: [arXiv:0709.2205] (2007)Google Scholar
  9. 9.
    Jayasumana, S., Hartley, R., Salzmann, M., Li, H., Harandi, M.: Kernel methods on the Riemannian manifold of symmetric positive definite matrices. In: CVPR, pp. 73–80 (2013)Google Scholar
  10. 10.
    Jayasumana, S., Hartley, R., Salzmann, M., Li, H., Harandi, M.: Optimizing over radial kernels on compact manifolds. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (June 2014)Google Scholar
  11. 11.
    Jhuang, H., Garrote, E., Yu, X., Khilnani, V., Poggio, T., Steele, A.D., Serre, T.: Automated home-cage behavioural phenotyping of mice. Nature Communications 1, 68 (2010)CrossRefGoogle Scholar
  12. 12.
    Kulis, B., Grauman, K.: Kernelized locality-sensitive hashing. IEEE Trans. Pattern Analysis and Machine Intelligence 34(6), 1092–1104 (2012)CrossRefGoogle Scholar
  13. 13.
    Micchelli, C.A., Xu, Y., Zhang, H.: Universal kernels. Journal of Machine Learning Research 7, 2651–2667 (2006)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Ojala, T., Pietikainen, M., Maenpaa, T.: Multiresolution gray-scale and rotation invariant texture classification with local binary patterns. IEEE Trans. Pattern Analysis and Machine Intelligence 24(7), 971–987 (2002)CrossRefGoogle Scholar
  15. 15.
    Scholkopf, B.: The kernel trick for distances. In: Proc. Advances in Neural Information Processing Systems (NIPS), pp. 301–307 (2001)Google Scholar
  16. 16.
    Schölkopf, B., Herbrich, R., Smola, A.J.: A generalized representer theorem. In: Helmbold, D.P., Williamson, B. (eds.) COLT/EuroCOLT 2001. LNCS (LNAI), vol. 2111, pp. 416–426. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press (2004)Google Scholar
  18. 18.
    Sim, T., Baker, S., Bsat, M.: The cmu pose, illumination, and expression database. IEEE Trans. Pattern Analysis and Machine Intelligence 25(12), 1615–1618 (2003)CrossRefGoogle Scholar
  19. 19.
    Steinwart, I., Christmann, A.: Support vector machines. Springer (2008)Google Scholar
  20. 20.
    Strehl, A., Ghosh, J., Mooney, R.: Impact of similarity measures on web-page clustering. In: AAAI Workshop on Artificial Intelligence for Web Search, pp. 58–64 (2000)Google Scholar
  21. 21.
    Subbarao, R., Meer, P.: Nonlinear mean shift over Riemannian manifolds. Int. Journal of Computer Vision 84(1), 1–20 (2009)CrossRefGoogle Scholar
  22. 22.
    Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE Trans. Pattern Analysis and Machine Intelligence 33(11), 2273–2286 (2011)CrossRefGoogle Scholar
  23. 23.
    Vemulapalli, R., Pillai, J.K., Chellappa, R.: Kernel learning for extrinsic classification of manifold features. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1782–1789 (2013)Google Scholar
  24. 24.
    Wolf, L., Shashua, A.: Learning over sets using kernel principal angles. Journal of Machine Learning Research 4, 913–931 (2003)MathSciNetGoogle Scholar
  25. 25.
    Yu, S., Tan, T., Huang, K., Jia, K., Wu, X.: A study on gait-based gender classification. IEEE Trans. Image Processing (TIP) 18(8), 1905–1910 (2009)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Zheng, S., Zhang, J., Huang, K., He, R., Tan, T.: Robust view transformation model for gait recognition. In: International Conference on Image Processing (ICIP), pp. 2073–2076 (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mehrtash T. Harandi
    • 1
    • 2
  • Mathieu Salzmann
    • 1
    • 2
  • Sadeep Jayasumana
    • 1
    • 2
  • Richard Hartley
    • 1
    • 2
  • Hongdong Li
    • 1
    • 2
  1. 1.Australian National UniversityCanberraAustralia
  2. 2.NICTACanberraAustralia

Personalised recommendations