Geometric Structures Induced by Deformations of the Legendre Transform
Abstract
1. Introduction
2. Preliminaries
3. Legendre Transform in Information Geometry
3.1. The Dual Structure of Statistical Manifolds
3.2. Dually Flat Geometry, Bregman Divergences, and the Legendre Transform
3.3. Divergences as a General Tool to Establish Geometries
3.4. Generalized Legendre Transforms as a Natural Way to Describe Curved Manifolds
4. Symplectic and Kähler Structures in Information Geometry
4.1. Establishing Dynamics on Phase Space
4.2. Symplectic Structure under the Deformed Legendre Transform
Rényi’s Symplectic 2-Form and Flow
4.3. Complexification of Statistical Manifolds
- (1)
- on ;
- (2)
- for some .
4.4. Complex Rényi Geometry under the Deformed Legendre Transform
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Complex Polarizations
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Morales, P.A.; Korbel, J.; Rosas, F.E. Geometric Structures Induced by Deformations of the Legendre Transform. Entropy 2023, 25, 678. https://doi.org/10.3390/e25040678
Morales PA, Korbel J, Rosas FE. Geometric Structures Induced by Deformations of the Legendre Transform. Entropy. 2023; 25(4):678. https://doi.org/10.3390/e25040678
Chicago/Turabian StyleMorales, Pablo A., Jan Korbel, and Fernando E. Rosas. 2023. "Geometric Structures Induced by Deformations of the Legendre Transform" Entropy 25, no. 4: 678. https://doi.org/10.3390/e25040678
APA StyleMorales, P. A., Korbel, J., & Rosas, F. E. (2023). Geometric Structures Induced by Deformations of the Legendre Transform. Entropy, 25(4), 678. https://doi.org/10.3390/e25040678