Conformal vector fields on statistical manifolds
DOI:
https://doi.org/10.33044/revuma.2118Abstract
Introducing the conformal vector fields on a statistical manifold, we present necessary and sufficient conditions for a vector field on a statistical manifold to be conformal. After presenting some examples, we classify the conformal vector fields on two famous statistical manifolds. Considering three statistical structures on the tangent bundle of a statistical manifold, we study the conditions under which the complete and horizontal lifts of a vector field can be conformal on these structures.
Downloads
References
M. T. K. Abbassi, N. Amri and C.-L. Bejan, Conformal vector fields and Ricci soliton structures on natural Riemann extensions, Mediterr. J. Math. 18 (2021), no. 2, Paper No. 55, 16 pp. MR 4215336.
S. Amari, Differential geometry of curved exponential families—curvatures and information loss, Ann. Statist. 10 (1982), no. 2, 357–385. MR 0653513.
S. Amari, Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, 28, Springer-Verlag, New York, 1985. MR 0788689.
S. Amari, Information geometry of the EM and em algorithms for neural networks, Neural Networks 8 (1995), no. 9, 1379–1408.
S. Amari and H. Nagaoka, Methods of Information Geometry, Translations of Mathematical Monographs, 191, American Mathematical Society, Providence, RI, 2000. MR 1800071.
K. A. Arwini and C. T. J. Dodson, Information Geometry: Near Randomness and Near Independence, Lecture Notes in Mathematics, 1953, Springer-Verlag, Berlin, 2008. MR 2450404.
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 (2020), no. 1, 120–135. MR 4066874.
C.-L. Bejan and Ş. Eken, Conformality on semi-Riemannian manifolds, Mediterr. J. Math. 13 (2016), no. 4, 2185–2198. MR 3530925.
M. Belkin, P. Niyogi and V. Sindhwani, Manifold regularization: a geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res. 7 (2006), 2399–2434. MR 2274444.
A. M. Blaga and M. Crasmareanu, Statistical structures in almost paracontact geometry, Bull. Iranian Math. Soc. 44 (2018), no. 6, 1407–1413. MR 3878399.
L. Cao, H. Sun and X. Wang, The geometric structures of the Weibull distribution manifold and the generalized exponential distribution manifold, Tamkang J. Math. 39 (2008), no. 1, 45–51. MR 2416179.
A. Caticha, The information geometry of space and time, https://arxiv.org/abs/gr-qc/0508108, 2005.
A. Caticha, Geometry from information geometry, https://arxiv.org/abs/1512.09076 [gr-qc], 2015.
S. Dutta and M. G. Genton, A non-Gaussian multivariate distribution with all lower-dimensional Gaussians and related families, J. Multivariate Anal. 132 (2014), 82–93. MR 3266261.
B. Efron, Defining the curvature of a statistical problem (with applications to second order efficiency), Ann. Statist. 3 (1975), no. 6, 1189–1242. MR 0428531.
R. A. Fisher, On the mathematical foundations of theoretical statistics, Philos. Trans. Roy. Soc. London 222 (1922), 309–368.
H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato and M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys. 117 (2017), 179–186. MR 3645840.
A. Gezer and L. Bilen, On infinitesimal conformal transformations with respect to the Cheeger–Gromoll metric, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 20 (2012), no. 1, 113–127. MR 2928413.
G. S. Hall and J. D. Steele, Conformal vector fields in general relativity, J. Math. Phys. 32 (1991), no. 7, 1847–1853. MR 1112714.
I. Hasegawa and K. Yamauchi, Conformally-projectively flat statistical structures on tangent bundles over statistical manifolds, in Differential Geometry and Its Applications, 239–251, World Sci. Publ., Hackensack, NJ, 2008. MR 2462797.
I. Hasegawa and K. Yamauchi, Conformal-projective flatness of tangent bundle with complete lift statistical structure, Differ. Geom. Dyn. Syst. 10 (2008), 148–158. MR 2390009.
S. Lauritzen, Statistical manifolds, in Differential Geometry in Statistical Inference, 163–216, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 10, Institute of Mathematical Statistics, Hayward, CA, 1987. MR 0932246.
H. Matsuzoe and J. Inoguchi, Statistical structures on tangent bundles, Appl. Sci. 5 (2003), no. 1, 55–75. MR 1950593.
P. J. Olver, Complex analysis and conformal mapping, 2022, 87 pages. https://www-users.cse.umn.edu/ olver/ln_/cml.pdf
E. Peyghan and A. Heydari, Conformal vector fields on tangent bundle of a Riemannian manifold, J. Math. Anal. Appl. 347 (2008), no. 1, 136–142. MR 2433831.
N. A. Pundeera, M. Ali, N. Ahmad and Z. Ahsan, Semiconformal symmetry—A new symmetry of the spacetime manifold of the general relativity, J. Math. Computer Sci. 20 (2020), no. 3, 241–254. http://dx.doi.org/10.22436/jmcs.020.03.07.
J. D. Qualls, Lectures on conformal field theory, https://arxiv.org/abs/1511.04074 [hep-th], 2015.
C. Radhakrishna Rao, Information and the accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc. 37 (1945), 81–91. MR 0015748.
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2) 10 (1958), 338–354. MR 0112152.
B. Ṣahin, Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems, Acta Appl. Math. 109 (2010), no. 3, 829–847. MR 2596178.
K. Sun and S. Marchand-Maillet, An information geometry of statistical manifold learning, in Proceedings of the 31st International Conference on Machine Learning ICML '14, 1–9, Proc. Mach. Learn. Res. (PMLR), 32, no. 2, 2014. https://proceedings.mlr.press/v32/suna14.html.
K. Yamauchi, On infinitesimal conformal transformations of the tangent bundles over Riemannian manifolds, Ann. Rep. Asahikawa Med. Coll. 15 (1994), 1–10. https://amcor.asahikawa-med.ac.jp/modules/xoonips/detail.php?id=K15-1.
K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Pure and Applied Mathematics, No. 16, Marcel Dekker, New York, 1973. MR 0350650.
M. Yuan, On the geometric structure of some statistical manifolds, Balkan J. Geom. Appl. 24 (2019), no. 2, 79–89. MR 3994264.
J. Zhang, A note on curvature of $alpha$-connections of a statistical manifold, Ann. Inst. Statist. Math. 59 (2007), no. 1, 161–170. MR 2396035.
J. Zhang, Nonparametric information geometry: from divergence function to referential-representational biduality on statistical manifolds, Entropy 15 (2013), no. 12, 5384–5418. MR 3147060.
J. Zhang and F. Li, Symplectic and Kähler structures on statistical manifolds induced from divergence functions, in Geometric Science of Information, 595–603, Lecture Notes in Comput. Sci., 8085, Springer, Heidelberg, 2013. MR 3126092.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Leila Samereh, Esmaeil Peyghan
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.
