Research Article
Year 2021,
Volume: 50 Issue: 4, 1140 - 1154, 06.08.2021
Abstract
References
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- [2] S. Amari, Information geometry of the EM and em algorithms for neural networks, Neural Networks, 8 (9), 1379–1408, 1995.
- [3] F. Asgari and H.R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo, II. Ser. 67 (2), 185–195, 2018.
- [4] V. Balan, E. Peyghan and E. Sharahi, Statistical structures on the tangent bundle of a statistical manifold with Sasaki metric, Hacet. J. Math. Stat. 49 (1), 120–135, 2020.
- [5] M. Belkin, P. Niyogi and V. Sindhwani, Manifold regularization: a geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res. 7, 2399– 2434, 2006.
- [6] L. Bilen and A. Gezer, Some results on Riemannian g-natural metrics generated by classical lifts on the tangent bundle, Eurasian Math. J. 8 (4), 18–34, 2017.
- [7] T. Fei and J. Zhang, Interaction of Codazzi couplings with (Para-)Kähler geometry, Result Math. 72 (4), 2037–2056, 2017.
- [8] S. Gudmundsson and E. Kappos, On the geometry of the tangent bundles, Expo. Math. 20, 1–41, 2002.
- [9] S. Ianus, Statistical manifolds and tangent bundles, Sci. Bull. Univ. Politechnica of Bucharest Ser. D, 56, 29–34, 1994.
- [10] S.L. Lauritzen, Statistical manifolds, In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser. 10, Inst. Math. Statist. Hayward California, 96–163, 1987.
- [11] H. Matsuzoe and J.I. Inoguchi, Statistical structures on tangent bundles, APPS. Appl.Sci. 5 (1), 55–57, 2003.
- [12] K. Nomizu and T. Sasaki, Affine Differential Geometry: Geometry of Affine Immersions, vol. 111 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1994.
- [13] L. Nourmohammadifar, E. Peyghan and S. Uddin, Geometry of almost Kenmotsu Hom-Lie algebras, Quaest. Math. DOI: 10.2989/16073606.2021.1886194.
- [14] C.R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc. 37, 81–91, 1945.
- [15] A. Schwenk-Schellschmidt and U. Simon, Codazzi-equivalent affine connections, Result Math. 56, 211–229, 2009.
- [16] K. Sun and S. Marchand-Maillet, An information geometry of statistical manifold learning, (Proceedings of the 31st International Conference on Machine Learning (ICML-14), 1–9, 2014.
- [17] K. Yano, and S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, Inc., New York 1973.
Year 2021,
Volume: 50 Issue: 4, 1140 - 1154, 06.08.2021
Abstract
Let $TM$ be a tangent bundle over a Riemannian manifold $M$ with a Riemannian metric $g$ and $TG$ be a tangent Lie group over a Lie group with a left-invariant metric $g$. The purpose of the paper is two folds. Firstly, we study statistical structures on the tangent bundle $TM$ equipped with two Riemannian $g$-natural metrics and lift connections. Secondly, we define a left-invariant complete lift connection on the tangent Lie group $TG$ equipped with metric $\tilde{g}$ introduced in [F. Asgari and H. R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo II. Series, 2018] and study statistical structures in this setting.
References
- [1] M.T.K. Abbassi and M. Sarih, On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Diff. Geom. Appl. 22, 19–47, 2005.
- [2] S. Amari, Information geometry of the EM and em algorithms for neural networks, Neural Networks, 8 (9), 1379–1408, 1995.
- [3] F. Asgari and H.R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo, II. Ser. 67 (2), 185–195, 2018.
- [4] V. Balan, E. Peyghan and E. Sharahi, Statistical structures on the tangent bundle of a statistical manifold with Sasaki metric, Hacet. J. Math. Stat. 49 (1), 120–135, 2020.
- [5] M. Belkin, P. Niyogi and V. Sindhwani, Manifold regularization: a geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res. 7, 2399– 2434, 2006.
- [6] L. Bilen and A. Gezer, Some results on Riemannian g-natural metrics generated by classical lifts on the tangent bundle, Eurasian Math. J. 8 (4), 18–34, 2017.
- [7] T. Fei and J. Zhang, Interaction of Codazzi couplings with (Para-)Kähler geometry, Result Math. 72 (4), 2037–2056, 2017.
- [8] S. Gudmundsson and E. Kappos, On the geometry of the tangent bundles, Expo. Math. 20, 1–41, 2002.
- [9] S. Ianus, Statistical manifolds and tangent bundles, Sci. Bull. Univ. Politechnica of Bucharest Ser. D, 56, 29–34, 1994.
- [10] S.L. Lauritzen, Statistical manifolds, In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser. 10, Inst. Math. Statist. Hayward California, 96–163, 1987.
- [11] H. Matsuzoe and J.I. Inoguchi, Statistical structures on tangent bundles, APPS. Appl.Sci. 5 (1), 55–57, 2003.
- [12] K. Nomizu and T. Sasaki, Affine Differential Geometry: Geometry of Affine Immersions, vol. 111 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1994.
- [13] L. Nourmohammadifar, E. Peyghan and S. Uddin, Geometry of almost Kenmotsu Hom-Lie algebras, Quaest. Math. DOI: 10.2989/16073606.2021.1886194.
- [14] C.R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc. 37, 81–91, 1945.
- [15] A. Schwenk-Schellschmidt and U. Simon, Codazzi-equivalent affine connections, Result Math. 56, 211–229, 2009.
- [16] K. Sun and S. Marchand-Maillet, An information geometry of statistical manifold learning, (Proceedings of the 31st International Conference on Machine Learning (ICML-14), 1–9, 2014.
- [17] K. Yano, and S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, Inc., New York 1973.
There are 17 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | August 6, 2021 |
| Published in Issue | Year 2021 Volume: 50 Issue: 4 |