# Geometric Structures Induced by Deformations of the Legendre Transform

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## 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)
- ${\partial}_{i,{j}^{\prime}}\mathcal{D}={\partial}_{{j}^{\prime},i}\mathcal{D}$ on $\mathcal{M}\times \mathcal{M}$;
- (2)
- ${\partial}_{i,j}\mathcal{D}+{\partial}_{{i}^{\prime},{j}^{\prime}}\mathcal{D}=\kappa \phantom{\rule{0.166667em}{0ex}}{\partial}_{i,{j}^{\prime}}\mathcal{D}$ for some $\kappa \in \mathbb{R}$.

#### 4.4. Complex Rényi Geometry under the Deformed Legendre Transform

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Complex Polarizations

## References

- Rockafellar, R.T. Convex Analysis; Princeton University Press: Princeton, NJ, USA, 1997; Volume 11. [Google Scholar]
- McDuff, D.; Salamon, D. Introduction to Symplectic Topology; Oxford University Press: Oxford, UK, 2017; Volume 27. [Google Scholar]
- Jackson, D.M.; Kempf, A.; Morales, A.H. A robust generalization of the Legendre transform for QFT. J. Phys. A Math. Theor.
**2017**, 50, 225201. [Google Scholar] [CrossRef] - Krupková, O.; Smetanová, D. Legendre transformation for regularizable Lagrangians in field theory. Lett. Math. Phys.
**2001**, 58, 189–204. [Google Scholar] [CrossRef] - Amari, S.I. Information Geometry and Its Applications; Springer: Berlin/Heidelberg, Germany, 2016; Volume 194. [Google Scholar]
- Amari, S.I. Information geometry on hierarchy of probability distributions. IEEE Trans. Inf. Theory
**2001**, 47, 1701–1711. [Google Scholar] [CrossRef] - Ohara, A. Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation. Eur. Phys. J. B
**2009**, 70, 15–28. [Google Scholar] [CrossRef] - Scarfone, A.M.; Matsuzoe, H.; Wada, T. Information geometry of κ-exponential families: Dually-flat, Hessian and Legendre structures. Entropy
**2018**, 20, 436. [Google Scholar] [CrossRef] - Wong, T.K.L. Logarithmic divergences from optimal transport and Rényi geometry. Inf. Geom.
**2018**, 1, 39–78. [Google Scholar] [CrossRef] - Morales, P.A.; Rosas, F.E. Generalization of the maximum entropy principle for curved statistical manifolds. Phys. Rev. Res.
**2021**, 3, 033216. [Google Scholar] [CrossRef] - Wong, T.K.L.; Zhang, J. Tsallis and Rényi Deformations Linked via a New λ-Duality. IEEE Trans. Inf. Theory
**2022**, 68, 5353–5373. [Google Scholar] [CrossRef] - Stéphan, J.M.; Inglis, S.; Fendley, P.; Melko, R.G. Geometric mutual information at classical critical points. Phys. Rev. Lett.
**2014**, 112, 127204. [Google Scholar] [CrossRef] - Stéphan, J.M. Shannon and Rényi mutual information in quantum critical spin chains. Phys. Rev. B
**2014**, 90, 045424. [Google Scholar] [CrossRef] - Dong, X. The Gravity Dual of Renyi Entropy. Nat. Commun.
**2016**, 7, 12472. [Google Scholar] [CrossRef] [PubMed] - Barrella, T.; Dong, X.; Hartnoll, S.A.; Martin, V.L. Holographic entanglement beyond classical gravity. J. High Energy Phys.
**2013**, 9, 109. [Google Scholar] [CrossRef] - Jizba, P.; Korbel, J. Maximum entropy principle in statistical inference: Case for non-Shannonian entropies. Phys. Rev. Lett.
**2019**, 122, 120601. [Google Scholar] [CrossRef] [PubMed] - Iaconis, J.; Inglis, S.; Kallin, A.B.; Melko, R.G. Detecting classical phase transitions with Renyi mutual information. Phys. Rev. B
**2013**, 87, 195134. [Google Scholar] [CrossRef] - Zaletel, M.P.; Bardarson, J.H.; Moore, J.E. Logarithmic Terms in Entanglement Entropies of 2D Quantum Critical Points and Shannon Entropies of Spin Chains. Phys. Rev. Lett.
**2011**, 107, 020402. [Google Scholar] [CrossRef] - Jizba, P.; Arimitsu, T. The world according to Rényi: Thermodynamics of multifractal systems. Ann. Phys.
**2004**, 312, 17–59. [Google Scholar] [CrossRef] - Jizba, P.; Arimitsu, T. Observability of Rényi’s Entropy. Phys. Rev. E
**2004**, 69, 026128. [Google Scholar] [CrossRef] - Morales, P.A.; Korbel, J.; Rosas, F.E. Thermodynamics of exponential Kolmogorov-Nagumo averages. arXiv
**2023**, arXiv:2302.06959. [Google Scholar] - Zia, R.K.; Redish, E.F.; McKay, S.R. Making sense of the Legendre transform. Am. J. Phys.
**2009**, 77, 614–622. [Google Scholar] [CrossRef] - Boyd, S.; Boyd, S.P.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Villani, C. Optimal Transport: Old and New; Springer: Berlin/Heidelberg, Germany, 2009; Volume 338. [Google Scholar]
- Amari, S.I. Information geometry. Jpn. J. Math
**2021**, 16, 1–48. [Google Scholar] [CrossRef] - Vitagliano, V.; Sotiriou, T.P.; Liberati, S. The dynamics of metric-affine gravity. Ann. Phys.
**2011**, 326, 1259–1273, Erratum in Ann. Phys.**2013**, 329, 186–187.. [Google Scholar] [CrossRef] - Vitagliano, V. The role of nonmetricity in metric-affine theories of gravity. Class. Quant. Grav.
**2014**, 31, 045006. [Google Scholar] [CrossRef] - Amari, S.I.; Ikeda, S.; Shimokawa, H. Information geometry of α-projection in mean-field approximation. In Recent Developments of Mean Field Approximation; Opper, M., Saad, D., Eds.; MIT Press: Cambridge, UK, 2000. [Google Scholar]
- Amari, S.I.; Cichocki, A. Information geometry of divergence functions. Bull. Pol. Acad. Sci. Tech. Sci.
**2010**, 58, 183–195. [Google Scholar] [CrossRef] - Eguchi, S. Second order efficiency of minimum contrast estimators in a curved exponential family. Ann. Stat.
**1983**, 11, 793–803. [Google Scholar] [CrossRef] - Matumoto, T. Any statistical manifold has a contrast function—On the C3-functions taking the minimum at the diagonal of the product manifold. Hiroshima Math. J
**1993**, 23, 327–332. [Google Scholar] [CrossRef] - Ay, N.; Amari, S.I. A novel approach to canonical divergences within information geometry. Entropy
**2015**, 17, 8111–8129. [Google Scholar] [CrossRef] - Liese, F.; Vajda, I. On Divergences and Informations in Statistics and Information Theory. IEEE Trans. Inf. Theory
**2006**, 52, 4394–4412. [Google Scholar] [CrossRef] - Amari, S.I. α-Divergence is Unique, Belonging to Both f-Divergence and Bregman Divergence Classes. IEEE Trans. Inf. Theor.
**2009**, 55, 4925–4931. [Google Scholar] [CrossRef] - Chentsov, N. Statistical Decision Rules and Optimal Inference; Transl. Math.; Monographs, American Mathematical Society: Providence, RI, USA, 1982. [Google Scholar]
- Ay, N.; Jost, J.; Vân Lê, H.; Schwachhöfer, L. Information geometry and sufficient statistics. Probab. Theory Relat. Fields
**2015**, 162, 327–364. [Google Scholar] [CrossRef] - Vân Lê, H. The uniqueness of the Fisher metric as information metric. Ann. Inst. Stat. Math.
**2017**, 69, 879–896. [Google Scholar] - Dowty, J.G. Chentsov’s theorem for exponential families. Inf. Geom.
**2018**, 1, 117–135. [Google Scholar] [CrossRef] - Cencov, N.N. Statistical Decision Rules and Optimal Inference; Number 53; American Mathematical Soc.: Providence, RI, USA, 2000. [Google Scholar]
- Amari, S.I. Differential geometry of curved exponential families-curvatures and information loss. Ann. Stat.
**1982**, 10, 357–385. [Google Scholar] [CrossRef] - Zhang, J. Divergence function, duality, and convex analysis. Neural Comput.
**2004**, 16, 159–195. [Google Scholar] [CrossRef] [PubMed] - Pal, S.; Wong, T.K.L. Exponentially concave functions and a new information geometry. Ann. Probab.
**2018**, 46, 1070–1113. [Google Scholar] [CrossRef] - Rényi, A. Selected Papers of Alfréd Rényi; Number 2 in Selected Papers of Alfréd Rényi, Akadémiai Kiadó; JSTOR: New York, NJ, USA, 1976. [Google Scholar]
- Valverde-Albacete, F.; Peláez-Moreno, C. The Case for Shifting the Rényi Entropy. Entropy
**2019**, 21, 46. [Google Scholar] [CrossRef] - Kochetov, E. SU (2) coherent-state path integral. J. Math. Phys.
**1995**, 36, 4667–4679. [Google Scholar] [CrossRef] - Brody, D.C.; Hughston, L.P. Geometric quantum mechanics. J. Geom. Phys.
**2001**, 38, 19–53. [Google Scholar] [CrossRef] - Gawȩdzki, K. Non-compact WZW conformal field theories. In New Symmetry Principles in Quantum Field Theory; Springer: Berlin/Heidelberg, Germany, 1992; pp. 247–274. [Google Scholar]
- Bellucci, S.; Nersessian, A. (Super) oscillator on CP
^{N}and a constant magnetic field. Phys. Rev. D**2005**, 71, 089901, Erratum in Phys. Rev. D**2003**, 67, 065013.. [Google Scholar] [CrossRef] - Woodhouse, N.M.J. Geometric Quantization; Oxford University Press: Oxford, UK, 1997. [Google Scholar]
- Bates, S.; Weinstein, A. Lectures on the Geometry of Quantization; American Mathematical Soc.: Providence, RI, USA, 1997; Volume 8. [Google Scholar]
- Arnold, V.I. Mathematical Methods of Classical Mechanics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 60. [Google Scholar]
- Zhang, J.; Li, F. Symplectic and Kähler structures on statistical manifolds induced from divergence functions. In Proceedings of the International Conference on Geometric Science of Information, Paris, France, 28–30 August 2013; Springer: Berlin/Heidelberg, Germany, 2013; pp. 595–603. [Google Scholar]
- Leok, M.; Zhang, J. Connecting information geometry and geometric mechanics. Entropy
**2017**, 19, 518. [Google Scholar] [CrossRef] - Candelas, P. Lectures on complex manifolds. In Superstrings and Grand Unification; 1988; Available online: https://inis.iaea.org/search/search.aspx?orig_q=RN:23060635 (accessed on 8 April 2023).
- Bouchard, V. Lectures on complex geometry, Calabi-Yau manifolds and toric geometry. arXiv
**2007**, arXiv:0702063. [Google Scholar] - Nakahara, M. Geometry, Topology and Physics; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Berndtsson, B. Convexity on the space of Kähler metrics. Ann. Fac. Des Sci. Toulouse Math.
**2013**, 22, 713–746. [Google Scholar] [CrossRef] - Khan, G.; Zhang, J. The Kähler geometry of certain optimal transport problems. Pure Appl. Anal.
**2020**, 2, 397–426. [Google Scholar] [CrossRef] - Zhang, J. Divergence functions and geometric structures they induce on a manifold. In Geometric Theory of Information; Springer: Berlin/Heidelberg, Germany, 2014; pp. 1–30. [Google Scholar]
- Zhang, W.M.; Feng, D.H.; Gilmore, R. Coherent states: Theory and some applications. Rev. Mod. Phys.
**1990**, 62, 867–927. [Google Scholar] [CrossRef] - Copinger, P.; Morales, P. Schwinger pair production in SL(2,$\mathbb{C}$) topologically nontrivial fields via non-Abelian worldline instantons. Phys. Rev. D
**2021**, 103, 036004. [Google Scholar] [CrossRef] - Kazinski, P.O. Stochastic deformation of a thermodynamic symplectic structure. Phys. Rev. E
**2009**, 79, 011105. [Google Scholar] [CrossRef] [PubMed] - Duval, C.; Horvath, Z.; Horvathy, P.A.; Martina, L.; Stichel, P. Berry phase correction to electron density in solids and ‘exotic’ dynamics. Mod. Phys. Lett. B
**2006**, 20, 373–378. [Google Scholar] [CrossRef] - Son, D.T.; Yamamoto, N. Berry Curvature, Triangle Anomalies, and the Chiral Magnetic Effect in Fermi Liquids. Phys. Rev. Lett.
**2012**, 109, 181602. [Google Scholar] [CrossRef]

**Figure 1.**A graphical representation of the Legendre transform and its deformations. (

**Left:**) While the standard Legendre transform acts on concave duals, the deformed one acts between C-concave functions. Each transform brings elements of one space to the other. Please note that while Section 2 presents the classical view of Legendre transforms acting over convex functions, the rest of this work follows Ref. [9] in focusing on concave functions. (

**Right:**) The symmetry that governs the algebraic relationships between convex dual functions and dual coordinates, which is mediated by the Legendre derivative operator ${\mathrm{D}}_{\mathrm{L}}$, which differs from the standard Euclidean gradient when the transform is deformed.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Morales, 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