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Entropy
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Information Geometry and Its Applications
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Information Geometry and Its Applications

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 8108

Special Issue Editors

Prof. Dr. Constantin Udriste

E-Mail Website
Guest Editor
Department of Mathematics-Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Splaiul Independentei 313, Sector 6, 060042 Bucharest, Romania
Interests: differential geometry; information geometry; optimizations on Riemannian manifolds; magnetic dynamical systems; geometric dynamics; multitime optimal control
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ionel Tevy

E-Mail Website
Guest Editor
Department of Mathematics-Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Splaiul Independentei 313, Sector 6, 060042 Bucharest, Romania
Interests: analysis; real functions; integrals; information geometry; optimization

Special Issue Information

Dear Colleagues,

Information geometry is a method of exploring the world of information by means of modern differential geometry.

The mathematical field of Information Geometry originated from the papers of C.R. Rao, who used Fisher information to define a Riemannian metric in spaces of probability distributions, and the papers of S. I. Amari, who showed that the differential-geometric structure of a statistical manifold can be derived from divergence functions, yielding a Riemannian metric and a pair of dually coupled affine connections.

The methods of Information Geometry have been applied to a wide variety of topics in physics, mathematical finance, biology and the neurosciences.

Now, there are many perspectives on information geometry as attested by the new MDPI journals, Springer journals, international conferences for “Geometric Sciences of Information”, books, etc. 

Topics: statistical manifolds and submanifolds, information geometry of space-time, Information geometry versus Riemannian geometry, dualistic structures of manifolds in information geometry, conjugate connections from divergence, dually flat spaces and canonical Bregman divergences, information geometry associated with a single-time Hamiltonian, information geometry associated with a multi-time Hamiltonian, dual Laplacians, applications of Information Geometry, stochastic information.

Prof. Dr. Constantin Udriste
Prof. Dr. Ionel Tevy
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • differential geometry
  • conjugate connections
  • dual metric-compatible parallel transport
  • information manifold
  • statistical manifold
  • curvature and flatness
  • dually flat manifolds
  • Hessian manifolds
  • exponential family
  • mixture family
  • statistical divergence
  • parameter divergence
  • separable divergence
  • divergence functions
  • Fisher–Rao distance
  • statistical invariance
  • Bayesian hypothesis testing
  • mixture clustering
  • α-embeddings
  • mixed parameterization
  • gauge freedom
  • statistical Roegenian economics
  • ideal income
  • Van der Waals income
  • economic partition function
  • Fisher–Rao metric
  • scalar curvature
  • geodesics
  • stochastic information

Published Papers (5 papers)

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Research

76 pages, 654 KiB  
Article
Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold
by Qi Feng and Wuchen Li
Entropy 2023, 25(5), 786; https://doi.org/10.3390/e25050786 - 11 May 2023
Cited by 2 | Viewed by 1208
Abstract
We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by [...] Read more.
We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric. Full article
(This article belongs to the Special Issue Information Geometry and Its Applications)
20 pages, 754 KiB  
Article
Geometric Structures Induced by Deformations of the Legendre Transform
by Pablo A. Morales, Jan Korbel and Fernando E. Rosas
Entropy 2023, 25(4), 678; https://0-doi-org.brum.beds.ac.uk/10.3390/e25040678 - 18 Apr 2023
Cited by 1 | Viewed by 1551
Abstract
The recent link discovered between generalized Legendre transforms and non-dually flat statistical manifolds suggests a fundamental reason behind the ubiquity of Rényi’s divergence and entropy in a wide range of physical phenomena. However, these early findings still provide little intuition on the nature [...] Read more.
The recent link discovered between generalized Legendre transforms and non-dually flat statistical manifolds suggests a fundamental reason behind the ubiquity of Rényi’s divergence and entropy in a wide range of physical phenomena. However, these early findings still provide little intuition on the nature of this relationship and its implications for physical systems. Here we shed new light on the Legendre transform by revealing the consequences of its deformation via symplectic geometry and complexification. These findings reveal a novel common framework that leads to a principled and unified understanding of physical systems that are not well-described by classic information-theoretic quantities. Full article
(This article belongs to the Special Issue Information Geometry and Its Applications)
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41 pages, 13660 KiB  
Article
A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions
by Frank Nielsen
Entropy 2023, 25(4), 654; https://0-doi-org.brum.beds.ac.uk/10.3390/e25040654 - 13 Apr 2023
Cited by 3 | Viewed by 2988
Abstract
We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We [...] Read more.
We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We consider experimentally the linear interpolation curves in the ordinary, natural, and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller’s isometric embedding of the Fisher–Rao d-variate normal manifold into the cone of (d+1)×(d+1) symmetric positive–definite matrices. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information–geometric properties of Calvo and Oller’s isometric embedding. Full article
(This article belongs to the Special Issue Information Geometry and Its Applications)
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9 pages, 270 KiB  
Article
A Dually Flat Embedding of Spacetime
by Jan Naudts
Entropy 2023, 25(4), 651; https://0-doi-org.brum.beds.ac.uk/10.3390/e25040651 - 13 Apr 2023
Cited by 1 | Viewed by 671
Abstract
A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The [...] Read more.
A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The present work explores their role in describing the geometry of spacetime. It is shown that the positive-definite metric with its flat 5-d connection can coexist with a pseudometric for which the connection is that of Levi–Civita. The 4-d geodesics are characterized by five conserved quantities, one of which can be chosen freely and is taken equal to zero in the present work. An explicit expression for the parallel transport operators is obtained. It is used to construct a pseudometric for spacetime by choosing an arbitrary possibly degenerate inner product in the tangent space of a reference point, for instance, that of Minkowski. By parallel transport, one obtains a pseudometric for spacetime, the metric connection of which extends to a 5-d connection with vanishing curvature tensor. The de Sitter space is considered as an example. Full article
(This article belongs to the Special Issue Information Geometry and Its Applications)
12 pages, 271 KiB  
Article
Information Geometry in Roegenian Economics
by Constantin Udriste and Ionel Tevy
Entropy 2022, 24(7), 932; https://0-doi-org.brum.beds.ac.uk/10.3390/e24070932 - 05 Jul 2022
Cited by 1 | Viewed by 974
Abstract
We characterise the geometry of the statistical Roegenian manifold that arises from the equilibrium distribution of an income of noninteracting identical economic actors. The main results for ideal income are included in three subsections: partition function in distribution, scalar curvature, and geodesics. Although [...] Read more.
We characterise the geometry of the statistical Roegenian manifold that arises from the equilibrium distribution of an income of noninteracting identical economic actors. The main results for ideal income are included in three subsections: partition function in distribution, scalar curvature, and geodesics. Although this system displays no phase transition, its analysis provides an enlightening contrast with the results of Van der Waals Income in Roegenian Economics, where we shall examine the geometry of the economic Van der Waals income, which does exhibit a “monetary policy as liquidity—income” transition. Here we focus on three subsections: canonical partition function, economic limit, and information geometry of the economic Van der Waals manifold. Full article
(This article belongs to the Special Issue Information Geometry and Its Applications)
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