Mixture Probabilistic Principal Geodesic Analysis

  • Youshan ZhangEmail author
  • Jiarui Xing
  • Miaomiao Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11846)


Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis (PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality. In this paper, we present a novel Gaussian latent variable model that provides a unique way to integrate multiple PGA models into a maximum-likelihood framework. This leads to a well-defined mixture model of probabilistic principal geodesic analysis (MPPGA) on sub-populations, where parameters of the principal subspaces are automatically estimated by employing an Expectation Maximization algorithm. We further develop a mixture Bayesian PGA (MBPGA) model that automatically reduces data dimensionality by suppressing irrelevant principal geodesics. We demonstrate the advantages of our model in the contexts of clustering and statistical shape analysis, using synthetic sphere data, real corpus callosum, and mandible data from human brain magnetic resonance (MR) and CT images.


  1. 1.
    Banerjee, M., Jian, B., Vemuri, B.C.: Robust Fréchet mean and PGA on riemannian manifolds with applications to neuroimaging. In: Niethammer, M., et al. (eds.) IPMI 2017. LNCS, vol. 10265, pp. 3–15. Springer, Cham (2017). Scholar
  2. 2.
    Bishop, C.M.: Bayesian PCA. In: Advances in Neural Information Processing Systems, pp. 382–388 (1999)Google Scholar
  3. 3.
    Bishop, C.M.: Pattern recognition and machine learning, pp. 500–600 (2006)Google Scholar
  4. 4.
    Chen, J., Liu, J.: Mixture principal component analysis models for process monitoring. Ind. Eng. Chem. Res. 38(4), 1478–1488 (1999)CrossRefGoogle Scholar
  5. 5.
    Chung, M.K., Qiu, A., Seo, S., Vorperian, H.K.: Unified heat kernel regression for diffusion, kernel smoothing and wavelets on manifolds and its application to mandible growth modeling in ct images. Med. Image Anal. 22(1), 63–76 (2015)CrossRefGoogle Scholar
  6. 6.
    Cootes, T.F., Taylor, C.J.: A mixture model for representing shape variation. Image Vis. Comput. 17(8), 567–573 (1999)CrossRefGoogle Scholar
  7. 7.
    Do Carmo, M.: Riemannian Geometry. Birkhauser (1992)Google Scholar
  8. 8.
    Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fletcher, P.T.: Geodesic regression and the theory of least squares on riemannian manifolds. Int. J. Comput. Vis. 105(2), 171–185 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8), 995–1005 (2004)CrossRefGoogle Scholar
  11. 11.
    Fletcher, P.T., Zhang, M.: Probabilistic geodesic models for regression and dimensionality reduction on riemannian manifolds. In: Turaga, P.K., Srivastava, A. (eds.) Riemannian Computing in Computer Vision, pp. 101–121. Springer, Cham (2016). Scholar
  12. 12.
    Jolliffe, I.T.: Principal component analysis and factor analysis. In: Jolliffe, I.T. (ed.) Principal Component Analysis. Springer Series in Statistics, pp. 115–128. Springer, New York (1986). Scholar
  13. 13.
    Kaufman, L., Rousseeuw, P.J.: Partitioning around medoids (program PAM). In: Finding Groups in Data: An Introduction to Cluster Analysis, pp. 68–125 (1990)Google Scholar
  14. 14.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ketchen, D.J., Shook, C.L.: The application of cluster analysis in strategic management research: an analysis and critique. Strat. Manag. J. 17(6), 441–458 (1996)CrossRefGoogle Scholar
  16. 16.
    Mardia, K.V., Jupp, P.E.: Directional Statistics, vol. 494. Wiley, Hoboken (2009)zbMATHGoogle Scholar
  17. 17.
    Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14(3), 333–340 (1962)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Roweis, S.T.: EM algorithms for PCA and SPCA. In: Advances in Neural Information Processing Systems, pp. 626–632 (1998)Google Scholar
  19. 19.
    Sommer, S., Lauze, F., Hauberg, S., Nielsen, M.: Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6316, pp. 43–56. Springer, Heidelberg (2010). Scholar
  20. 20.
    Sommer, S., Lauze, F., Nielsen, M.: Optimization over geodesics for exact principal geodesic analysis. Adv. Comput. Math. 40(2), 283–313 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tibshirani, R., Walther, G., Hastie, T.: Estimating the number of clusters in a data set via the gap statistic. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 63(2), 411–423 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tipping, M.E., Bishop, C.M.: Mixtures of probabilistic principal component analyzers. Neural Comput. 11(2), 443–482 (1999)CrossRefGoogle Scholar
  23. 23.
    Tipping, M.E., Bishop, C.M.: Probabilistic principal component analysis. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 61(3), 611–622 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on grassmann and stiefel manifolds for image and video-based recognition. IEEE Trans. Pattern Anal. Mach. Intell. 33(11), 2273–2286 (2011)CrossRefGoogle Scholar
  25. 25.
    Tuzel, O., Porikli, F., Meer, P.: Pedestrian detection via classification on riemannian manifolds. IEEE Trans. Pattern Anal. Mach. Intell. 30(10), 1713–1727 (2008)CrossRefGoogle Scholar
  26. 26.
    Zhang, M., Fletcher, P.T.: Probabilistic principal geodesic analysis. In: Advances in Neural Information Processing Systems, pp. 1178–1186 (2013)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Computer Science and EngineeringLehigh UniversityBethlehemUSA
  2. 2.Electrical and Computer EngineeringUniversity of VirginiaCharlottesvilleUSA
  3. 3.Computer ScienceUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations