Skip to main content
SpringerLink
Log in
Book cover

International Conference on Geometric Science of Information

GSI 2019: Geometric Science of Information pp 739–748Cite as

Curvature of the Manifold of Fixed-Rank Positive-Semidefinite Matrices Endowed with the Bures–Wasserstein Metric

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11712))

Abstract

We consider the manifold of rank-p positive-semidefinite matrices of size n, seen as a quotient of the set of full-rank n-by-p matrices by the orthogonal group in dimension p. The resulting distance coincides with the Wasserstein distance between centered degenerate Gaussian distributions. We obtain expressions for the Riemannian curvature tensor and the sectional curvature of the manifold. We also provide tangent vectors spanning planes associated with the extreme values of the sectional curvature.

This work was supported by (i) the Fonds de la Recherche Scientifique – FNRS and the Fonds Wetenschappelijk Onderzoek – Vlaanderen under EOS Project no 30468160, (ii) “Communauté française de Belgique - Actions de Recherche Concertées” (contract ARC 14/19-060), (iii) the WBI-World Excellence Fellowship.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
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

Learn about institutional subscriptions

References

  1. Absil, P.A., Gousenbourger, P.-Y., Striewski, P., Wirth, B.: Differentiable piecewise-Bézier surfaces on Riemannian manifolds. SIAM J. Imaging Sci. 9(4), 1788–1828 (2016). http://dx.doi.org/10.1137/16M1057978

    Article  MathSciNet  Google Scholar 

  2. Afsari, B., Tron, R., Vidal, R.: On the convergence of gradient descent for finding the Riemannian center of mass. SIAM J. Control Optim. 51(3), 2230–2260 (2013). https://doi.org/10.1137/12086282X

    Article  MathSciNet  MATH  Google Scholar 

  3. Bhatia, R., Jain, T., Lim, Y.: On the Bures-Wasserstein distance between positive definite matrices. Expositiones Mathematicae (2018). https://doi.org/10.1016/j.exmath.2018.01.002

  4. Bonnabel, S.: Stochastic gradient descent on Riemannian manifolds. IEEE Trans. Autom. Control 58(9), 2217–2229 (2013). https://doi.org/10.1109/TAC.2013.2254619

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhatia, R., Rosenthal, P.: How and why to solve the operator equation AX- XB= Y. Bull. Lond. Math. Soc. 29(1), 1–21 (1997)

    Article  MathSciNet  Google Scholar 

  6. Bonnabel, S., Sepulchre, R.: Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31(3), 1055–1070 (2009). https://doi.org/10.1137/080731347

    Article  MathSciNet  MATH  Google Scholar 

  7. Dittmann, J.: On the Riemannian metric on the space of density matrices. Rep. Math. Phys. 36(2–3), 309–315 (1995). https://doi.org/10.1016/0034-4877(96)83627-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Gelbrich, G.: On a formula for the \(L^2\) Wasserstein metric between measures on Euclidean and Hilbert spaces. Math. Nachr. 147(1), 185–203 (1990). https://doi.org/10.1002/mana.19901470121

    Article  MathSciNet  MATH  Google Scholar 

  9. Gousenbourger, P.-Y., et al.: Piecewise-Bézier \(C^1\) smoothing on manifolds with application to wind field estimation. In: Proceedings of the 25th European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN), pp. 305–310 (2017)

    Google Scholar 

  10. Journée, M., Bach, F., Absil, P.-A., Sepulchre, R.: Low-rank optimization on the cone of positive semidefinite matrices. SIAM J. Opti. 20(5), 2327–2351 (2010). https://doi.org/10.1137/080731359

    Article  MathSciNet  MATH  Google Scholar 

  11. Kacem, A., Daoudi, M., Amor, B.B, Berretti, S., Alvarez-Paiva, J.C.: A Novel geometric framework on gram matrix trajectories for human behavior understanding. IEEE Trans. Pattern Anal. Mach. Intell. (T-PAMI) (2018). https://doi.org/10.1109/tpami.2018.2872564

  12. Li, X.-B., Burkowski, F.J.: Conformational transitions and principal geodesic analysis on the positive semidefinite matrix manifold. In: Basu, M., Pan, Y., Wang, J. (eds.) ISBRA 2014. LNCS, vol. 8492, pp. 334–345. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08171-7_30

    Chapter  Google Scholar 

  13. Massart, E., Absil, P.-A.: Quotient geometry of the manifold of fixed-rank positive-semidefinite matrices. Technical report UCL-INMA-2018.06, UCLouvain, November 2018, Preprint. http://sites.uclouvain.be/absil/2018.06

  14. Meyer, G., Bonnabel, S., Sepulchre, R.: Regression on fixed-rank positive semidefinite matrices: a Riemannian approach. J. Mach. Learn. Res. 12, 593–625 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Massart, E., Gousenbourger, P.-Y., Son, N.T., Stykel, T., Absil, P.-A.: Interpolation on the manifold of fixed-rank positive-semidefinite matrices for parametric model order reduction: preliminary results. In: Proceedings of the 27th European Symposium on Artifical Neural Networks, Computational Intelligence and Machine Learning (ESANN2019), pp. 281–286 (2019)

    Google Scholar 

  16. Marchand, M., Huang, W., Browet, A., Van Dooren, P., Gallivan, K.A.: A Riemannian optimization approach for role model extraction. In: Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems, pp. 58–64 (2016)

    Google Scholar 

  17. Malagò, L., Montrucchio, L., Pistone, G.: Wasserstein Riemannian geometry of Gaussian densities. Inf. Geom. 1(2), 137–179 (2018). https://doi.org/10.1007/s41884-018-0014-4

    Article  MATH  Google Scholar 

  18. Mishra, B., Meyer, G., Sepulchre, R.: Low-rank optimization for distance matrix completion. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 4455–4460 (2011). https://doi.org/10.1109/CDC.2011.6160810

  19. O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13(4), 459–469 (1966). https://doi.org/10.1307/mmj/1028999604

    Article  MathSciNet  MATH  Google Scholar 

  20. O’Neill, B.: Semi-Riemannian geometry. In: Pure and Applied Mathematics, vol. 103. Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York (1983)

    Google Scholar 

  21. Samir, C., Absil, P.-A., Srivastava, A., Klassen, E.: A gradient-descent method for curve fitting on Riemannian manifolds. Found. Comput. Math. 12(1), 49–73 (2012)

    Article  MathSciNet  Google Scholar 

  22. Smith, S.T.: Covariance, subspace, and intrinsic Cramér-Rao bounds. IEEE Trans. Signal Process 53(5), 1610–1630 (2005)

    Article  MathSciNet  Google Scholar 

  23. Takatsu, A.: Wasserstein geometry of Gaussian measures. Osaka J. Math. 48(4), 1005–1026 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Vandereycken, B., Absil, P.-A., Vandewalle, S.: Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank. In: IEEE/SP 15th Workshop on Statistical Signal Processing, pp. 389–392 (2009). https://doi.org/10.1109/SSP.2009.5278558

  25. Vandereycken, B., Absil, P.-A., Vandewalle, S.: A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank. IMA J. Numer. Anal. 33(2), 481–514 (2013). https://doi.org/10.1093/imanum/drs006

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ICTEAM, UCLouvain, Louvain-la-Neuve, Belgium

    Estelle Massart, Julien M. Hendrickx & P.-A. Absil

  2. CISE Resident Scholar at Boston University, Boston, USA

    Julien M. Hendrickx

Authors
  1. Estelle Massart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julien M. Hendrickx
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P.-A. Absil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Estelle Massart .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories, Inc., Tokyo, Japan

    Frank Nielsen

  2. Thales, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Massart, E., Hendrickx, J.M., Absil, PA. (2019). Curvature of the Manifold of Fixed-Rank Positive-Semidefinite Matrices Endowed with the Bures–Wasserstein Metric. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_77

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26980-7_77

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26979-1

  • Online ISBN: 978-3-030-26980-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation