Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures
In this paper, we present a review of recent developments on the κ -deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the κ -formalism which are obtained starting from three, not equivalent,...
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MDPI AG
2018-06-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/20/6/436 |
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| author | Antonio M. Scarfone Hiroshi Matsuzoe Tatsuaki Wada |
| author_facet | Antonio M. Scarfone Hiroshi Matsuzoe Tatsuaki Wada |
| author_sort | Antonio M. Scarfone |
| collection | DOAJ |
| description | In this paper, we present a review of recent developments on the κ -deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the κ -formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the κ -deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the κ -Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the κ → 0 limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the κ -escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of κ -thermodynamics in the picture of the information geometry. |
| first_indexed | 2024-12-10T07:04:12Z |
| format | Article |
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| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-12-10T07:04:12Z |
| publishDate | 2018-06-01 |
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| series | Entropy |
| spelling | doaj.art-e6747f6c87694c2da0947d8ae2c8514b2022-12-22T01:58:13ZengMDPI AGEntropy1099-43002018-06-0120643610.3390/e20060436e20060436Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre StructuresAntonio M. Scarfone0Hiroshi Matsuzoe1Tatsuaki Wada2Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Politecnico di Torino, 10129 Torino, ItalyDepartment of Computer Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, JapanRegion of Electrical and Electronic Systems Engineering, Ibaraki University, Nakanarusawa-cho, Hitachi 316-8511, JapanIn this paper, we present a review of recent developments on the κ -deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the κ -formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the κ -deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the κ -Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the κ → 0 limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the κ -escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of κ -thermodynamics in the picture of the information geometry.http://www.mdpi.com/1099-4300/20/6/436κ-generalized statistical mechanicsinformation geometrydually-flat geometryHessian geometryLegendre structuredivergence functions |
| spellingShingle | Antonio M. Scarfone Hiroshi Matsuzoe Tatsuaki Wada Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures Entropy κ-generalized statistical mechanics information geometry dually-flat geometry Hessian geometry Legendre structure divergence functions |
| title | Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures |
| title_full | Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures |
| title_fullStr | Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures |
| title_full_unstemmed | Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures |
| title_short | Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures |
| title_sort | information geometry of κ exponential families dually flat hessian and legendre structures |
| topic | κ-generalized statistical mechanics information geometry dually-flat geometry Hessian geometry Legendre structure divergence functions |
| url | http://www.mdpi.com/1099-4300/20/6/436 |
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