Wasserstein Riemannian Geometry on Statistical Manifold

Year 2020, Volume: 13 Issue: 2, 144 - 151, 15.10.2020

Carlos Ogouyandjou , Nestor Wadagnı

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.                                                                                                                                             ..............................................

Keywords

Statistical manifold, Riemannian metric, Otto metric, α-connections, Wasserstein Riemannian space, flatness

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).