Wasserstein Riemannian Geometry on Statistical Manifold
Year 2020,
, 144 - 151, 15.10.2020
Carlos Ogouyandjou
Nestor Wadagnı
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
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Year 2020,
, 144 - 151, 15.10.2020
Carlos Ogouyandjou
Nestor Wadagnı
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).
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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).
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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.
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- Reference 1 3 Villani, C.: Topics in Optimal Transportation. Graduate studies in Mathematics 58, Providence, RI: Ameri. Math. Soc. (2003).