Skip to main content
SpringerLink
Log in

Deterministic Walks and Quasi-Subgradient Methods for the Karcher Mean on NPC Spaces

  • Conference paper
Geometric Science of Information (GSI 2013)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8085))

Included in the following conference series:

Abstract

In this article we provide several new approximation methods for the Karcher or Fréchet mean on NPC spaces. We provide a non-stochastic version of Sturm’s law of large numbers, i.e. a weighted deterministic walk for the Karcher mean. Then we extend certain subgradient methods existing in the case of Riemannian manifolds to the case of non-differentiable case of NPC spaces for the Karcher mean. These methods not only provide new intrinsic algorithms for computing the Karcher mean in NPC spaces but also new theoretical results and characterizations for the Karcher mean.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnaudon, M., Barbaresco, F., Le, Y.: Medians and means in Riemannian geometry: existence, uniqueness and computation, http://arxiv.org/abs/1111.3120

  2. Barachant, A., Bonnet, S., Congedo, M., Jutten, C.: Riemannian geometry applied to BCI classification (preprint)

    Google Scholar 

  3. Bini, D., Iannazzo, B.: Computing the Karcher mean of symmetric positive definite matrices (preprint)

    Google Scholar 

  4. Es-Sahib, A., Heinich, H.: Barycentre canonique pour un espace métrique à courbure négative. Séminaire de Probabilités (Strasbourg) 33, 355–370 (1999)

    MathSciNet  Google Scholar 

  5. Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Mathematical Programming 62, 261–275 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cartan, E.: Groupes simples clos et ouverts et gomtrie Riemannienne. J. Math. Pure. Appl. 8, 1–34 (1929)

    MathSciNet  MATH  Google Scholar 

  7. Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser, Basel (2006)

    Google Scholar 

  8. Holbrook, J.: No dice: a determinic approach to the Cartan centroid. To Appear in J. Ramanujan Math. Soc.

    Google Scholar 

  9. Larotonda, G.: Nonpositive curvature: a geometrical approach to Hilbert-Schmidt operators. Differential Geom. Appl. 25(6), 679–700 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lawson, J., Lim, Y.: A general framework for extending means to higher orders. Colloq. Math. 113, 191–221 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lawson, J., Lee, H., Lim, Y.: Weighted geometric means. Forum Math. (to appear)

    Google Scholar 

  12. Lim, Y., Pálfia, M.: The matrix power means and the Karcher mean. J. Func. Anal. 262(4), 1498–1514 (2012)

    Article  MATH  Google Scholar 

  13. Lim, Y., Pálfia, M.: A deterministic approach for the Karcher mean on Hadamard spaces (submitted)

    Google Scholar 

  14. Lim, Y., Pálfia, M.: Weighted random walks and no dice approach for the least squares mean on Hadamard spaces (preprint, 2013)

    Google Scholar 

  15. Moakher, M.: On the averaging of symmetric positive-definite tensors. Journal of Elasticity 82, 273–296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nedić, A., Bertsekas, D.: Convergence Rate of Incremental Subgradient Algorithms. In: Uryasev, S., Pardalos, P.M. (eds.) Stochastic Optimization: Algorithms and Applications, pp. 263–304. Kluwer Academic Publishers (2000)

    Google Scholar 

  17. Pálfia, M.: Means in metric spaces and the center of mass, J. Math. Anal. Appl. 381, 383–391 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pálfia, M.: Quasi-subgradient methods for the Karcher mean on NPC spaces (preprint, 2013)

    Google Scholar 

  19. Pennec, X.: Statistical computing on manifolds: From riemannian geometry to computational anatomy. In: Nielsen, F. (ed.) ETVC 2008. LNCS, vol. 5416, pp. 347–386. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  20. Sturm, K.-T.: Probability measures on metric spaces of nonpositive curvature. In: Auscher, P., et al. (eds.) Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces. Contemp. Math., vol. 338, Amer. Math. Soc. (AMS), Providence (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science, Sungkyunkwan University, Suwon, 440-746, Korea

    Miklós Pálfia

Authors
  1. Miklós Pálfia
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc, Tokyo, Japan

    Frank Nielsen

  2. Thales Land Air Systems, 91470, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pálfia, M. (2013). Deterministic Walks and Quasi-Subgradient Methods for the Karcher Mean on NPC Spaces. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_90

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40020-9_90

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40019-3

  • Online ISBN: 978-3-642-40020-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation