Skip to main content
SpringerLink
Log in

Advances in Geometric Statistics for Manifold Dimension Reduction

  • Chapter
  • First Online:
Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

Geometric statistics aim at shifting the classical paradigm for inference from points in a Euclidean space to objects living in a non-linear space, in a consistent way with the underlying geometric structure considered. In this chapter, we illustrate some recent advances of geometric statistics for dimension reduction in manifolds. Beyond the mean value (the best zero-dimensional summary statistics of our data), we want to estimate higher dimensional approximation spaces fitting our data. We first define a family of natural parametric geometric subspaces in manifolds that generalize the now classical geodesic subspaces: barycentric subspaces are implicitly defined as the locus of weighted means of k + 1 reference points with positive or negative weights summing up to one. Depending on the definition of the mean, we obtain the Fréchet, Karcher or Exponential Barycentric subspaces (FBS/KBS/EBS). The completion of the EBS, called the affine span of the points in a manifold is the most interesting notion as it defines complete sub-(pseudo)-spheres in constant curvature spaces. Barycentric subspaces can be characterized very similarly to the Euclidean case by the singular value decomposition of a certain matrix or by the diagonalization of the covariance and the Gram matrices. This shows that they are stratified spaces that are locally manifolds of dimension k at regular points. Barycentric subspaces can naturally be nested by defining an ordered series of reference points in the manifold. This allows the construction of inductive forward or backward properly nested sequences of subspaces approximating data points. These flags of barycentric subspaces generalize the sequence of nested linear subspaces (flags) appearing in the classical Principal Component Analysis. We propose a criterion on the space of flags, the accumulated unexplained variance (AUV), whose optimization exactly lead to the PCA decomposition in Euclidean spaces. This procedure is called barycentric subspace analysis (BSA). We illustrate the power of barycentric subspaces in the context of cardiac imaging with the estimation, analysis and reconstruction of cardiac motion from sequences of images.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Page 259: “It is not certain that such an element exists nor that it is unique.”

  2. 2.

    Note III on normal spaces with negative or null Riemannian curvature, p. 267.

  3. 3.

    Appliquons au point origine O les différents déplacements définis par les transformations de γ. Le groupe γ étant clos, nous obtenons ainsi une variété fermée V (qui peut se réduire à un point). Or, dans un espace de Riemann sans point singulier à distance finie, simplement connexe, a courbure negative ou nulle, on peut trouver, étant donnés des points en nombre fini, un point fixe invariant par tous les déplacements qui échangent entre eux les points donnés: c’est le point pour lequel la somme des carrés des distances au point donné est minima [4, p. 267]. Cette propriété est encore vraie si, au lieu d’un nombre fini de points, on en a une infinité formant une variété fermée: nous arrivons donc à la conclusion que le groupe γ qui laisse évidemment invariante la variété V, laisse invariant un point fixe de l’espace, il fait done partie du groupe des rotations (isométriques) autour de ce point. Mais ce groupe est homologue à g dans le groupe adjoint continu, ce qui démontre le théoreme.

  4. 4.

    p-jets are equivalent classes of functions up to order p. Thus, a p-jet specifies the Taylor expansion of a smooth function up to order p. Non-local jets, or multijets, generalize subspaces of the tangent spaces to higher differential orders with multiple base points.

References

  1. Afsari, B.: Riemannian L p center of mass: existence, uniqueness, and convexity. Proc. Am. Math. Soc. 139(2), 655–673 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  3. Buser, P., Karcher, H.: Gromov’s Almost Flat Manifolds. Number 81 in Astérisque. Société mathématique de France (1981)

    Google Scholar 

  4. Cartan, E.: Leçons sur la géométrie des espaces de Riemann. Gauthier-Villars, Paris (1928)

    MATH  Google Scholar 

  5. Cartan, E.: Groupes simples clos et ouverts et géométrie riemannienne. J. Math. Pures Appl. 9e série(tome 8), 1–34 (1929)

    Google Scholar 

  6. Damon, J., Marron, J.S.: Backwards principal component analysis and principal nested relations. J. Math. Imaging Vision 50(1–2), 107–114 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Darling, R.W.R.: Geometrically intrinsic nonlinear recursive filters II: foundations (1998). arXiv:math/9809029

    Google Scholar 

  8. Ding, C., Zhou, D., He, X., Zha, H.: R1-PCA: Rotational invariant L1-norm principal component analysis for robust subspace factorization. In: Proceedings of the 23rd International Conference on Machine Learning, ICML ’06, pp. 281–288. ACM, New York (2006)

    Google Scholar 

  9. Dryden, I., Mardia, K.: Theoretical and distributional aspects of shape analysis. In: Probability Measures on Groups, X (Oberwolfach, 1990), pp. 95–116, Plenum, New York (1991)

    Google Scholar 

  10. Edelman, A., Arias, T., Smith, S.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Emery, M., Mokobodzki, G.: Sur le barycentre d’une probabilité dans une variété. In: Séminaire de Probabilités XXV, vol. 1485, pp. 220–233. Springer, Berlin (1991)

    Google Scholar 

  12. Feragen, A., Owen, M., Petersen, J., Wille, M.M.W., Thomsen, L.H., Dirksen, A., de Bruijne, M.: Tree-space statistics and approximations for large-scale analysis of anatomical trees. In: Gee, J.C., Joshi, S., Pohl, K.M., Wells, W.M., Zöllei, L. (eds.) Information Processing in Medical Imaging, pp. 74–85. Springer, Berlin (2013)

    Chapter  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. Fréchet, M.: Valeurs moyennes attachées a un triangle aléatoire. La Revue Scientifique, Fascicule 10, 475–482 (1943)

    MATH  Google Scholar 

  15. Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincare 10, 215–310 (1948)

    MathSciNet  MATH  Google Scholar 

  16. Gramkow, C.: On averaging rotations. Int. J. Comput. Vis. 42(1–2), 7–16 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grenander, U., Miller, M., Srivastava, A.: Hilbert-Schmidt lower bounds for estimators on matrix Lie groups for ATR. IEEE Trans. Pattern Anal. Mach. Intell. 20(8), 790–802 (1998)

    Article  Google Scholar 

  18. Grove, K., Karcher, H.: How to conjugate C1-close group actions. Math. Z. 132(1), 11–20 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huckemann, S., Ziezold, H.: Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Adv. Appl. Probab. 38(2), 299–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions. Stat. Sin. 20, 1–100 (2010)

    MATH  Google Scholar 

  21. Jung, S., Dryden, I.L., Marron, J.S.: Analysis of principal nested spheres. Biometrika 99(3), 551–568 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  23. Karcher, H.: Riemannian center of mass and so called Karcher mean (2014). arXiv:1407.2087

    Google Scholar 

  24. Kendall, D.: A survey of the statistical theory of shape (with discussion). Stat. Sci. 4, 87–120 (1989)

    Article  MATH  Google Scholar 

  25. Kendall, W.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. Lond. Math. Soc. 61(2), 371–406 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kwak., N.: Principal component analysis based on L1-norm maximization. IEEE Trans. Pattern Anal. Mach. Intell. 30(9), 1672–1680 (2008)

    Google Scholar 

  27. Le, H.: Locating Fréchet means with application to shape spaces. Adv. Appl. Probab. 33, 324–338 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Le, H.: Estimation of Riemannian barycenters. LMS J. Comput. Math. 7, 193–200 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Le, H., Kendall, D.: The Riemannian structure of Euclidean shape space: a novel environment for statistics. Ann. Stat. 21, 1225–1271 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Leporé, N., Brun, C., Chou, Y.-Y., Lee, A., Barysheva, M., Pennec, X., McMahon, K., Meredith, M., De Zubicaray, G., Wright, M., Toga, A.W., Thompson, P.: Best individual template selection from deformation tensor minimization. In: Proceedings of the 2008 IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI’08), Paris, pp. 460–463 (2008)

    Google Scholar 

  31. Lorenzi, M., Pennec, X.: Geodesics, Parallel transport and one-parameter subgroups for diffeomorphic image registration. Int. J. Comput. Vis. 105(2), 111–127 (2013)

    MATH  Google Scholar 

  32. Marron, J.S., Alonso, A.M.: Overview of object oriented data analysis. Biom. J. 56(5), 732–753 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mcleod, K., Sermesant, M., Beerbaum, P., Pennec, X.: Spatio-temporal tensor decomposition of a polyaffine motion model for a better analysis of pathological left ventricular dynamics. IEEE Trans. Med. Imaging 34(7), 1562–1675 (2015)

    Article  Google Scholar 

  34. Moakher, M.: Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24(1), 1–16 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Oller, J., Corcuera, J.: Intrinsic analysis of statistical estimation. Ann. Stat. 23(5), 1562–1581 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pennec, X.: Probabilities and statistics on Riemannian manifolds: basic tools for geometric measurements. In: Cetin, A.E., Akarun, L., Ertuzun, A., Gurcan, M.N., Yardimci, Y. (eds.) Proceedings of Nonlinear Signal and Image Processing (NSIP’99), vol. 1, pp. 194–198. IEEE-EURASIP, Antalya (1999)

    Google Scholar 

  37. 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, 2004

    Google Scholar 

  38. Pennec, X.: Sample-limited L p barycentric subspace analysis on constant curvature spaces. In: Geometric Sciences of Information (GSI 2017), vol. 10589, pp. 20–28. Springer, Berlin (2017)

    Google Scholar 

  39. Pennec, X: Barycentric subspace analysis on manifolds. Ann. Stat. 46(6A), 2711–2746 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pennec, X., Arsigny, V.: Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups. In: Barbaresco, F., Mishra, A., Nielsen, F. (eds.) Matrix Information Geometry, pp. 123–168. Springer, Berlin (2012)

    Google Scholar 

  41. Pennec, X., Lorenzi, M.: Beyond Riemannian Geometry The affine connection setting for transformation groups chapter 5. In: Pennec, S.S.X., Fletcher, T. (eds.) Riemannian Geometric Statistics in Medical Image Analysis. Elsevier, Amsterdam (2019)

    Google Scholar 

  42. Pennec, X., Guttmann, C.R., Thirion, J.-P.: Feature-based registration of medical images: estimation and validation of the pose accuracy. In: Proceedings of First International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI’98). LNCS, vol. 1496, pp. 1107–1114. Springer, Berlin (1998)

    Google Scholar 

  43. 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, 2004

    Google Scholar 

  44. Rohé, M.-M., Sermesant, M., Pennec, X.: Barycentric subspace analysis: a new symmetric group-wise paradigm for cardiac motion tracking. In: 19th International Conference on Medical Image Computing and Computer Assisted Intervention, MICCAI 2016. Lecture Notes in Computer Science, vol. 9902, pp. 300–307, Athens (2016)

    Google Scholar 

  45. Rohé, M.-M., Sermesant, M., Pennec, X.: Low-dimensional representation of cardiac motion using barycentric subspaces: a new group-wise paradigm for estimation, analysis, and reconstruction. Med. Image Anal. 45, 1–12 (2018)

    Article  Google Scholar 

  46. Small, C.: The Statistical Theory of Shapes. Springer Series in Statistics. Springer, Berlin (1996)

    Google Scholar 

  47. Sommer, S.: Horizontal Dimensionality Reduction and Iterated Frame Bundle Development. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 76–83. Springer, Berlin (2013)

    Chapter  Google Scholar 

  48. Sommer, S., Lauze, F., Nielsen, M.: Optimization over geodesics for exact principal geodesic analysis. Adv. Comput. Math. 40(2), 283–313 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Tobon-Gomez, C., De Craene, M., Mcleod, K., Tautz, L., Shi, W., Hennemuth, A., Prakosa, A., Wang, H., Carr-White, G., Kapetanakis, S., Lutz, A., Rasche, V., Schaeffter, T., Butakoff, C., Friman, O., Mansi, T., Sermesant, M., Zhuang, X., Ourselin, S., Peitgen, H.O., Pennec, X., Razavi, R., Rueckert, D., Frangi, A.F., Rhode, K.: Benchmarking framework for myocardial tracking and deformation algorithms: an open access database. Med. Image Anal. 17(6), 632–648 (2013)

    Article  Google Scholar 

  50. Weyenberg, G.S.: Statistics in the Billera–Holmes–Vogtmann treespace. PhD thesis, University of Kentucky, 2015

    Google Scholar 

  51. Yang, L.: Riemannian median and its estimation. LMS J. Comput. Math. 13, 461–479 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yang, L.: Medians of probability measures in Riemannian manifolds and applications to radar target detection. PhD thesis, Poitier University, 2011

    Google Scholar 

  53. Zhai, H.: Principal component analysis in phylogenetic tree space. PhD thesis, University of North Carolina at Chapel Hill, 2016.

    Google Scholar 

  54. Ziezold, H.: On Expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In: Kožešnik, J. (ed.) Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the 1974 European Meeting of Statisticians, vol. 7A, pp. 591–602. Springer, Netherlands (1977)

    Chapter  Google Scholar 

Download references

Acknowledgements

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant G-Statistics No 786854).

Author information

Authors and Affiliations

  1. Universtié Côte d’Azur, Nice, France

    Xavier Pennec

  2. Inria, Sophia-Antipolis, France

    Xavier Pennec

Authors
  1. Xavier Pennec
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xavier Pennec .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Pennec, X. (2020). Advances in Geometric Statistics for Manifold Dimension Reduction. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_11

Download citation

Publish with us

Policies and ethics

Search

Navigation