Clustering Symmetric Positive Definite Matrices on the Riemannian Manifolds

  • Ligang ZhengEmail author
  • Guoping Qiu
  • Jiwu Huang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10111)


Using structured features such as symmetric positive definite (SPD) matrices to encode visual information has been found to be effective in computer vision. Traditional pattern recognition methods developed in the Euclidean space are not suitable for directly processing SPD matrices because they lie in Riemannian manifolds of negative curvature. The main contribution of this paper is the development of a novel framework, termed Riemannian Competitive Learning (RCL), for SPD matrices clustering. In this framework, we introduce a conscious competition mechanism and develop a robust algorithm termed Riemannian Frequency Sensitive Competitive Learning (rFSCL). Compared with existing methods, rFSCL has three distinctive advantages. Firstly, rFSCL inherits the online nature of competitive learning making it capable of handling very large data sets. Secondly, rFSCL inherits the advantage of conscious competitive learning which means that it is less sensitive to the initial values of the cluster centers and that all clusters are fully utilized without the “dead unit” problem associated with many clustering algorithms. Thirdly, as an intrinsic Riemannian clustering method, rFSCL operates along the geodesic on the manifold and the algorithms is completely independent of the choice of local coordinate systems. Extensive experiments show its superior performance compared with other state of the art SPD matrices clustering methods.


Riemannian Manifold Diffusion Tensor Imaging Geodesic Distance Competitive Learning Symmetric Positive Definite Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Part of this paper is supported by NSFC (61332012, 61300205), Shenzhen R&D Program (JCYJ20160328144421330, GJHZ20140418191518323).


  1. 1.
    Goh, A., Vidal, R.: Clustering and dimensionality reduction on Riemannian manifolds. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2008)Google Scholar
  2. 2.
    Chiang, M.C., Dutton, R.A., Hayashi, K.M., Lopez, O.L., Aizenstein, H.J., Toga, A.W., Becker, J.T., Thompson, P.M.: \(3d\) pattern of brain atrophy in HIV/AIDS visualized using tensor-based morphometry. Neuroimage 34, 44–60 (2007)CrossRefGoogle Scholar
  3. 3.
    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). doi: 10.1007/11744047_45 CrossRefGoogle Scholar
  4. 4.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Fast and simple calculus on tensors in the log-Euclidean framework. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 115–122. Springer, Heidelberg (2005). doi: 10.1007/11566465_15 CrossRefGoogle Scholar
  5. 5.
    Cherian, A., Sra, S., Banerjee, A., Papanikolopoulos, N.: Jensen-bregman logdet divergence with application to efficient similarity search for covariance matrices. IEEE Trans. Pattern Anal. Mach. Intell. 35, 2161–2174 (2013)CrossRefGoogle Scholar
  6. 6.
    Chaudhry, R., Ivanov, Y.: Fast approximate nearest neighbor methods for non-euclidean manifolds with applications to human activity analysis in videos. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6312, pp. 735–748. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15552-9_53 CrossRefGoogle Scholar
  7. 7.
    Malcolm, J., Rathi, Y., Tannenbaum, A.: A graph cut approach to image segmentation in tensor space. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–8 (2007)Google Scholar
  8. 8.
    Zheng, L., Lei, Y., Qiu, G., Huang, J.: Near-duplicate image detection in a visually salient Riemannian space. IEEE Trans. Inf. Forensics Secur. 7, 1578–1593 (2012)CrossRefGoogle Scholar
  9. 9.
    Alavi, A., Wiliem, A., Zhao, K., Lovell, B., Sanderson, C.: Random projections on manifolds of symmetric positive definite matrices for image classification. In: 2014 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 301–308 (2014)Google Scholar
  10. 10.
    Zhang, S., Kasiviswanathan, S., Yuen, P., Harandi, M.: Online dictionary learning on symmetric positive definite manifolds with vision applications. In: Twenty-Ninth AAAI Conference on Artificial Intelligence (AAAI 2015), pp. 3165–3173 (2015)Google Scholar
  11. 11.
    Bridson, M.R.: Metric Spaces of Non-Positive Curvature. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Tyagi, A., Davis, J.W.: A recursive filter for linear systems on riemannian manifolds. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)Google Scholar
  13. 13.
    Dryden, I., Koloydenko, A., Zhou, D.: Non-euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. Appl. Stat. 3, 1102–1123 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jayasumana, S., Hartley, R., Salzmann, M., Li, H., Harandi, M.: Kernel methods on the Riemannian manifold of symmetric positive definite matrices. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 73–80 (2013)Google Scholar
  15. 15.
    Salehian, H., Cheng, G., Vemuri, B., Ho, J.: Recursive estimation of the stein center of SPD matrices and its applications. In: IEEE International Conference on Computer Vision (ICCV), pp. 1793–1800 (2013)Google Scholar
  16. 16.
    Rathi, Y., Tannenbaum, A., Michailovich, O.: Segmenting images on the tensor manifold. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–8 (2007)Google Scholar
  17. 17.
    Sivalingam, R., Morellas, V., Boley, D., Papanikolopoulos, N.: Metric learning for semi-supervised clustering of region covariance descriptors. In: Third ACM/IEEE International Conference on Distributed Smart Cameras, pp. 1–8 (2009)Google Scholar
  18. 18.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66, 41–66 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  20. 20.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sra, S.: Positive definite matrices and the s-divergence (2013).
  22. 22.
    Zhao, K., Alavi, A., Wiliem, A., Lovell, B.C.: Efficient clustering on Riemannian manifolds: a kernelised random projection approach. Pattern Recogn. 51, 333–345 (2016)CrossRefGoogle Scholar
  23. 23.
    Hidot, S., Saint-Jean, C.: An expectation-maximization algorithm for the Wishart mixture model: application to movement clustering. Pattern Recogn. Lett. 31, 2318–2324 (2010)CrossRefGoogle Scholar
  24. 24.
    Cherian, A., Morellas, V., Papanikolopoulos, N., Bedros, S.J.: Dirichlet process mixture models on symmetric positive definite matrices for appearance clustering in video surveillance applications. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 3417–3424 (2011)Google Scholar
  25. 25.
    Hiai, F., Petz, D.: Riemannian metrics on positive definite matrices related to means. Linear Algebra Appl. 430, 3105–3130 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Frstner, W., Moonen, B.: A metric for covariance matrices. Technical report, Stuttgart University (1999)Google Scholar
  27. 27.
    Du, K.L.: Clustering: a neural network approach. Neural Netw. 23, 89–107 (2010)CrossRefGoogle Scholar
  28. 28.
    Ahalt, S.C., Krishnamurthy, A.K., Chen, P., Melton, D.E.: Competitive learning algorithms for vector quantization. Neural Netw. 3, 277–290 (1990)CrossRefGoogle Scholar
  29. 29.
    Banerjee, A., Ghosh, J.: Frequency-sensitive competitive learning for scalable balanced clustering on high-dimensional hyperspheres. IEEE Trans. Neural Netw. 15, 702–719 (2004)CrossRefGoogle Scholar
  30. 30.
    Galanopoulos, A.S., Moses, R.L., Ahalt, S.C.: Diffusion approximation of frequency sensitive competitive learning. IEEE Trans. Neural Netw. 8, 1026–1030 (1997)CrossRefGoogle Scholar
  31. 31.
    Qiu, G., Duana, J., Finlaysonb, G.D.: Learning to display high dynamic range images. Pattern Recogn. 40, 2641–2655 (2007)CrossRefGoogle Scholar
  32. 32.
    Basser, P.J., Mattiello, J., LeBihan, D.: Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. Ser. B 103(3), 247–254 (1994)CrossRefGoogle Scholar
  33. 33.
    Caputo, B., Hayman, E., Mallikarjuna, P.: Class-specific material categorisation. In: Proceedings of the Tenth IEEE International Conference on Computer Vision-ICCV 2005, vol. 2, pp. 1597–1604. IEEE Computer Society, Washington, DC (2005)Google Scholar
  34. 34.
    Dollár, P.: Piotr’s image and video matlab toolbox (pmt) (2013).
  35. 35.
    Leibe, B., Schiele, B.: Analyzing appearance and contour based methods for object categorization. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. II-409–II-415 (2003)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Guangzhou UniversityGuangzhouChina
  2. 2.University of NottinghamNottinghamUK
  3. 3.Shenzhen UniversityShenzhenChina

Personalised recommendations