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Wasserstein Riemannian Geometry on Statistical Manifold

Year 2020, , 144 - 151, 15.10.2020
https://doi.org/10.36890/iejg.689702

Abstract

In this paper, we study some geometric properties of statistical manifold equipped with the Riemannian Otto metric which is related to the L 2 -Wasserstein distance of optimal mass transport. We construct some α -connections on such manifold and we prove that the proposed connections are torsion-free and coincide with the Levi-Civita connection when α = 0 . In addition, the exponentialy families and the mixture families are shown to be respectively (1) -flat and (−1) -flat.                                                                                                                                             ..............................................

Supporting Institution

CEA-SMA (IMSP) University of Abomey-Calavi, Benin

Thanks

The authors would like to thank the CEA-SMA (IMSP) University of Abomey-Calavi, Benin for their support.

References

  • Reference 1 Amari, S-I., Barndorff-Nielsen O. E., Kass R. E., Lauritzen,S. L., Rao C. R.,: Differential geometry in statistical inference. Lecture notes-monograph series Shanti S., Series Editor vol. 10. (1987).
  • Reference 2 Amari, S., Nagaoko, H.,: Methods of information geometry. American Mathematical Soc. (2007).
  • Reference 3 De Souza, E.,: Tensor Calculus for Engineers and Physicists. Springer (2016).
  • Reference 4 De Giorgi, E.,: New problems on minimizing movements, Boundary Value Problems for PDE and Applications, pp. 81-98, Masson, Paris (1993).
  • Reference 5 Ambrosio, L., Gigli, N., Savar\'e, G.,: Gradient flows in metric space and in the space of probability measures, Lectures in Mathematics ETH, second ed., Birkhäuser Verlag, Besel (2008).
  • Reference 6 Do Carmo, M.,: Riemannian Geometry, Birkhauser Inc., Boston (1992). Reference 7 Gbaguidi Amoussou, A., Djibril Moussa, F., Ogouyandjou, C., Diop, M. A.,: New connections on the fiber-bundle of generalized statistical manifolds. Balkan Society of Geometers, Geometry Balkan Press 23-32 (2019).
  • Reference 8 Kantorovich, L.,: On the translocation of masses, C. R. Acad. Sci. URSS (N.S) 37, 199-201 (1942).
  • Reference 9 Lott, J.,: Some geometric calculation on Wasserstein space. Commun. Math. Phys. 277, 423-437 (2008). Reference 10 Olkin, I., Pukelsheim, F.: The distance between two random vectors with given dispersion matrices. Linear Algebra Appl. 48, 257–263 (1982). https://doi.org/10.1016/0024-3795(82)90112-4.
  • Reference 1 1 Rao, C.R.,: Information and accuracy attainable in the estimation of statistical parameter Bull. Calcutta. Math. Soc. 37, 81-91 (1945).
  • Reference 1 2 De Souza, D., Vigelis R., Cavalcante C.,: Geometry induced by a generalization of Rényi divergence. Entropy 18(11), 407 (2016).
  • Reference 1 3 Villani, C.: Topics in Optimal Transportation. Graduate studies in Mathematics 58, Providence, RI: Ameri. Math. Soc. (2003).
Year 2020, , 144 - 151, 15.10.2020
https://doi.org/10.36890/iejg.689702

Abstract

References

  • Reference 1 Amari, S-I., Barndorff-Nielsen O. E., Kass R. E., Lauritzen,S. L., Rao C. R.,: Differential geometry in statistical inference. Lecture notes-monograph series Shanti S., Series Editor vol. 10. (1987).
  • Reference 2 Amari, S., Nagaoko, H.,: Methods of information geometry. American Mathematical Soc. (2007).
  • Reference 3 De Souza, E.,: Tensor Calculus for Engineers and Physicists. Springer (2016).
  • Reference 4 De Giorgi, E.,: New problems on minimizing movements, Boundary Value Problems for PDE and Applications, pp. 81-98, Masson, Paris (1993).
  • Reference 5 Ambrosio, L., Gigli, N., Savar\'e, G.,: Gradient flows in metric space and in the space of probability measures, Lectures in Mathematics ETH, second ed., Birkhäuser Verlag, Besel (2008).
  • Reference 6 Do Carmo, M.,: Riemannian Geometry, Birkhauser Inc., Boston (1992). Reference 7 Gbaguidi Amoussou, A., Djibril Moussa, F., Ogouyandjou, C., Diop, M. A.,: New connections on the fiber-bundle of generalized statistical manifolds. Balkan Society of Geometers, Geometry Balkan Press 23-32 (2019).
  • Reference 8 Kantorovich, L.,: On the translocation of masses, C. R. Acad. Sci. URSS (N.S) 37, 199-201 (1942).
  • Reference 9 Lott, J.,: Some geometric calculation on Wasserstein space. Commun. Math. Phys. 277, 423-437 (2008). Reference 10 Olkin, I., Pukelsheim, F.: The distance between two random vectors with given dispersion matrices. Linear Algebra Appl. 48, 257–263 (1982). https://doi.org/10.1016/0024-3795(82)90112-4.
  • Reference 1 1 Rao, C.R.,: Information and accuracy attainable in the estimation of statistical parameter Bull. Calcutta. Math. Soc. 37, 81-91 (1945).
  • Reference 1 2 De Souza, D., Vigelis R., Cavalcante C.,: Geometry induced by a generalization of Rényi divergence. Entropy 18(11), 407 (2016).
  • Reference 1 3 Villani, C.: Topics in Optimal Transportation. Graduate studies in Mathematics 58, Providence, RI: Ameri. Math. Soc. (2003).
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Carlos Ogouyandjou 0000-0001-7115-7724

Nestor Wadagnı This is me 0000-0001-8757-176X

Publication Date October 15, 2020
Acceptance Date August 16, 2020
Published in Issue Year 2020

Cite

APA Ogouyandjou, C., & Wadagnı, N. (2020). Wasserstein Riemannian Geometry on Statistical Manifold. International Electronic Journal of Geometry, 13(2), 144-151. https://doi.org/10.36890/iejg.689702
AMA Ogouyandjou C, Wadagnı N. Wasserstein Riemannian Geometry on Statistical Manifold. Int. Electron. J. Geom. October 2020;13(2):144-151. doi:10.36890/iejg.689702
Chicago Ogouyandjou, Carlos, and Nestor Wadagnı. “Wasserstein Riemannian Geometry on Statistical Manifold”. International Electronic Journal of Geometry 13, no. 2 (October 2020): 144-51. https://doi.org/10.36890/iejg.689702.
EndNote Ogouyandjou C, Wadagnı N (October 1, 2020) Wasserstein Riemannian Geometry on Statistical Manifold. International Electronic Journal of Geometry 13 2 144–151.
IEEE C. Ogouyandjou and N. Wadagnı, “Wasserstein Riemannian Geometry on Statistical Manifold”, Int. Electron. J. Geom., vol. 13, no. 2, pp. 144–151, 2020, doi: 10.36890/iejg.689702.
ISNAD Ogouyandjou, Carlos - Wadagnı, Nestor. “Wasserstein Riemannian Geometry on Statistical Manifold”. International Electronic Journal of Geometry 13/2 (October 2020), 144-151. https://doi.org/10.36890/iejg.689702.
JAMA Ogouyandjou C, Wadagnı N. Wasserstein Riemannian Geometry on Statistical Manifold. Int. Electron. J. Geom. 2020;13:144–151.
MLA Ogouyandjou, Carlos and Nestor Wadagnı. “Wasserstein Riemannian Geometry on Statistical Manifold”. International Electronic Journal of Geometry, vol. 13, no. 2, 2020, pp. 144-51, doi:10.36890/iejg.689702.
Vancouver Ogouyandjou C, Wadagnı N. Wasserstein Riemannian Geometry on Statistical Manifold. Int. Electron. J. Geom. 2020;13(2):144-51.