From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices

  • Mehrtash T. Harandi
  • Mathieu Salzmann
  • Richard Hartley
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8690)


Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices –especially of high-dimensional ones– comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.


Riemannian geometry SPD manifold Grassmann manifold dimensionality reduction visual recognition 

Supplementary material

978-3-319-10605-2_2_MOESM1_ESM.pdf (567 kb)
Electronic Supplementary Material (PDF 568 KB)


  1. 1.
    Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  2. 2.
    Caseiro, R., Henriques, J.F., Martins, P., Batista, J.: Semi-intrinsic mean shift on riemannian manifolds. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part I. LNCS, vol. 7572, pp. 342–355. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Cherian, A., Sra, S., Banerjee, A., Papanikolopoulos, N.: Jensen-bregman logdet divergence with application to efficient similarity search for covariance matrices. IEEE Transactions on Pattern Analysis and Machine Intelligence 35(9), 2161–2174 (2013)CrossRefGoogle Scholar
  4. 4.
    Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging 23(8), 995–1005 (2004)CrossRefGoogle Scholar
  5. 5.
    Goh, A., Vidal, R.: Clustering and dimensionality reduction on riemannian manifolds. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–7. IEEE (2008)Google Scholar
  6. 6.
    Harandi, M.T., Sanderson, C., Hartley, R., Lovell, B.C.: Sparse coding and dictionary learning for symmetric positive definite matrices: A kernel approach. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part II. LNCS, vol. 7573, pp. 216–229. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Hussein, M.E., Torki, M., Gowayyed, M.A., El-Saban, M.: Human action recognition using a temporal hierarchy of covariance descriptors on 3d joint locations. In: Proc. Int. Joint Conference on Artificial Intelligence, IJCAI (2013)Google Scholar
  8. 8.
    Jayasumana, S., Hartley, R., Salzmann, M., Li, H., Harandi, M.: Kernel methods on the riemannian manifold of symmetric positive definite matrices. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (June 2013)Google Scholar
  9. 9.
    Jung, S., Dryden, I.L., Marron, J.: Analysis of principal nested spheres. Biometrika 99(3), 551–568 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lee, T.S.: Image representation using 2d Gabor wavelets. IEEE Transactions on Pattern Analysis and Machine Intelligence 18(10), 959–971 (1996)CrossRefGoogle Scholar
  11. 11.
    Li, H., Jiang, T., Zhang, K.: Efficient and robust feature extraction by maximum margin criterion. IEEE Transactions on Neural Networks 17(1), 157–165 (2006)CrossRefGoogle Scholar
  12. 12.
    Liao, Z., Rock, J., Wang, Y., Forsyth, D.: Non-parametric filtering for geometric detail extraction and material representation. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE (2013)Google Scholar
  13. 13.
    Lowe, D.G.: Distinctive image features from scale-invariant keypoints. IJCV 60(2), 91–110 (2004)CrossRefGoogle Scholar
  14. 14.
    Müller, M., Röder, T., Clausen, M., Eberhardt, B., Krüger, B., Weber, A.: Documentation: Mocap database HDM05. Tech. Rep. CG-2007-2, Universität Bonn (2007)Google Scholar
  15. 15.
    Pang, Y., Yuan, Y., Li, X.: Gabor-based region covariance matrices for face recognition. IEEE Transactions on Circuits and Systems for Video Technology 18(7), 989–993 (2008)CrossRefGoogle Scholar
  16. 16.
    Pennec, X., Fillard, P., Ayache, N.: A riemannian framework for tensor computing. Int. Journal of Computer Vision (IJCV) 66(1), 41–66 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Phillips, P.J., Moon, H., Rizvi, S.A., Rauss, P.J.: The feret evaluation methodology for face-recognition algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(10), 1090–1104 (2000)CrossRefGoogle Scholar
  18. 18.
    Sanin, A., Sanderson, C., Harandi, M., Lovell, B.: Spatio-temporal covariance descriptors for action and gesture recognition. In: IEEE Workshop on Applications of Computer Vision (WACV), pp. 103–110 (2013)Google Scholar
  19. 19.
    Sra, S.: A new metric on the manifold of kernel matrices with application to matrix geometric means. In: Proc. Advances in Neural Information Processing Systems (NIPS), pp. 144–152 (2012)Google Scholar
  20. 20.
    Tuzel, O., Porikli, F., Meer, P.: Region covariance: A fast descriptor for detection and classification. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 589–600. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Tuzel, O., Porikli, F., Meer, P.: Pedestrian detection via classification on riemannian manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence 30(10), 1713–1727 (2008)CrossRefGoogle Scholar
  22. 22.
    Wang, R., Guo, H., Davis, L.S., Dai, Q.: Covariance discriminative learning: A natural and efficient approach to image set classification. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2496–2503. IEEE (2012)Google Scholar
  23. 23.
    Wright, J., Yang, A.Y., Ganesh, A., Sastry, S.S., Ma, Y.: Robust face recognition via sparse representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 31(2), 210–227 (2009)CrossRefGoogle Scholar
  24. 24.
    Yan, S., Xu, D., Zhang, B., Zhang, H.J., Yang, Q., Lin, S.: Graph embedding and extensions: a general framework for dimensionality reduction. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(1), 40–51 (2007)CrossRefGoogle Scholar
  25. 25.
    Yang, M., Zhang, L.: Gabor feature based sparse representation for face recognition with Gabor occlusion dictionary. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part VI. LNCS, vol. 6316, pp. 448–461. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mehrtash T. Harandi
    • 1
    • 2
  • Mathieu Salzmann
    • 1
    • 2
  • Richard Hartley
    • 1
    • 2
  1. 1.Australian National UniversityCanberraAustralia
  2. 2.NICTACanberraAustralia

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