Advertisement

Barycentric Subspaces and Affine Spans in Manifolds

  • Xavier PennecEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)

Abstract

This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a “Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backward analysis and they are not symmetric in the parametrization of the subspaces. We propose in this paper a new and more general type of family of subspaces in manifolds: barycentric subspaces are implicitly defined as the locus of points which are weighted means of \(k+1\) reference points. Depending on the generalization of the mean that we use, we obtain the Fréchet/Karcher barycentric subspaces (FBS/KBS) or the affine span (with exponential barycenter). This definition restores the full symmetry between all parameters of the subspaces, contrarily to the geodesic subspaces which intrinsically privilege one point. We show that this definition defines locally a submanifold of dimension k and that it generalizes in some sense geodesic subspaces. Like PGA, barycentric subspaces allow the construction of a forward nested sequence of subspaces which contains the Fréchet mean. However, the definition also allows the construction of backward nested sequence which may not contain the mean. As this definition relies on points and do not explicitly refer to tangent vectors, it can be extended to non Riemannian geodesic spaces. For instance, principal subspaces may naturally span over several strata in stratified spaces, which is not the case with more classical generalizations of PCA.

Keywords

Riemannian Manifold Tangent Space Tangent Vector Nest Sequence Riemannian Curvature Tensor 
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.

References

  1. 1.
    Afsari, B.: Riemannian \(l^p\) center of mass: existence, uniqueness, and convexity. Proc. AMS 180(2), 655–673 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Brewin, L.: Riemann normal coordinate expansions using cadabra. Class. Quantum Gravity 26(17), 175017 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Damon, J., Marron, J.S.: Backwards principal component analysis and principal nested relations. J. Math. Imaging Vis. 50(1–2), 107–114 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fletcher, P., Lu, C., Pizer, S., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8), 995–1005 (2004)CrossRefGoogle Scholar
  5. 5.
    Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’Institut Henri Poincaré 10, 215–310 (1948)zbMATHGoogle Scholar
  6. 6.
    Gorban, A.N., Zinovyev, A.Y.: Principal graphs and manifolds. In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Chap. 2, pp. 28–59 (2009)Google Scholar
  7. 7.
    Jung, S., Dryden, I.L., Marron, J.S.: Analysis of principal nested spheres. Biometrika 99(3), 551–568 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Karcher, H.: Riemannian Center of Mass and so called Karcher mean, July 2014. arXiv:1407.2087 [math]
  10. 10.
    Kendall, W.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. Lond. Math. Soc. 61(2), 371–406 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Le, H.: Estimation of Riemannian barycenters. LMS J. Comput. Math 7, 193–200 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006). A preliminary appeared as INRIA RR-5093, January 2004MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006). A preliminary version appeared as INRIA Research. Report 5255, (July 2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Huckemann, A.M.S., Hotz, T.: Intrinsic shape analysis: geodesic principal component analysis for Riemannian manifolds modulo Lie group actions. Statistica Sin. 20, 1–100 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sommer, S.: Horizontal dimensionality reduction and iterated frame bundle development. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2013. LNCS, vol. 8085, pp. 76–83. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  16. 16.
    Sommer, S., Lauze, F., Nielsen, M.: Optimization over geodesics for exact principal geodesic analysis. Adv. Comput. Math. 40(2), 283–313 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yang, L.: Medians of probability measures in Riemannian manifolds and applications to radar target detection. Ph.D. thesis, Poitier University, December 2011Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Inria Sophia-Antipolis and Côte d’Azur University (UCA)Sophia AntipolisFrance

Personalised recommendations