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SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday

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

Deadline for manuscript submissions: 30 April 2026 | Viewed by 4879

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Special Issue Editors

Dr. Antonio M. Scarfone

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Guest Editor

Complex Systems Institute, Consiglio Nazionale delle Ricerche, 7-00185 Roma, Italy
Interests: non-extensive statistical mechanics; nonlinear Fokker–Planck equations; information geometry; nonlinear Schrödinger equation; quantum groups and quantum algebras; complex systems
Special Issues, Collections and Topics in MDPI journals

Dr. Tatsuaki Wada

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Guest Editor

Region of Electrical and Electronic Systems Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan
Interests: SUURI engineering; information geometry; non-extensive statistical mechanics; thermodynamics; electronic circuit theory; modified general relativity; cosmology

Special Issue Information

Dear Colleagues,

After the work of Rao, who first proposed the Fisher matrix as a Riemannian metric, the modern development of Information Geometry theory is largely due to SUURI engineer Professor Shun'ichi Amari, whose work has had an outstanding global impact among the mathematician and statistician communities. Information Geometry is a powerful formalism that applies concepts from differential geometry to the study of probability distributions and the understanding of curved statistical models. The framework of α-geometry helps us understand different statistical problems like Bayesian inference and optimization methods. It does this by introducing the ideas of α-divergence and dual connections, which are now important tools in statistical physics, machine learning, and optimization.

He has also been a key figure in the development of neural networks between the 1970s and 1980s, helping to establish a mathematical framework for analyzing artificial and biological neural networks. Today, with about 400 peer-reviewed articles, the ideas of SUURI engineer Professor Shun'ichi Amari remain fundamental in understanding the mathematical structures underlying data, learning, and intelligence. SUURI is a Japanese word consisting of two Kanji characters: SUU (numbers, or mathematical objects) and RI (reason, theory, or laws). Although this Japanese word is usually translated as “mathematics” or “applied physics”, it is more fairly related to the method of using mathematics to understand phenomena and problems.

In honor of SUURI engineer Professor Shun'ichi Amari’s 90th birthday, it is a pleasure to present this Special Issue, which aims to gather original research papers and high-quality reviews in Information Geometry and related fields. Works focusing on the successes and future challenges that Information Geometry aims to obtain through its mathematical formalism, theoretical foundations, and applications, especially in machine learning and information theory, are welcome. Contributions that seek to synthesize the potential applications of Information Geometry to statistical physics, as well as present novel results and prospects for its applications, are also encouraged.

Dr. Antonio M. Scarfone
Dr. Tatsuaki Wada
Guest Editors

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Keywords

information geometry
α-geometry
information theory
machine learning
neural network
statistics inference
geometro-thermodynamics

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Published Papers (5 papers)

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Research

42 pages, 447 KB

Open AccessArticle

An Approach to Fisher-Rao Metric for Infinite Dimensional Non-Parametric Information Geometry

by Bing Cheng and Howell Tong

Entropy 2026, 28(4), 374; https://doi.org/10.3390/e28040374 (registering DOI) - 25 Mar 2026

Abstract

Non-parametric information geometry has long faced an “intractability barrier”: in the infinite-dimensional setting, the Fisher–Rao metric is a weak Riemannian metric functional that lacks a bounded inverse, rendering classical optimization and estimation techniques computationally inaccessible. This paper resolves this barrier by building the [...] Read more.

L_{0}^{Φ} (P_{f})

(the Pistone–Sempi manifold), which provides the necessary exponential integrability for score functions and a rigorous Fréchet differentiability for the Kullback–Leibler divergence. We introduce a novel Structural Decomposition of the Tangent Space (

T_{f} M = S \oplus S^{⊥}

), where the infinite-dimensional space is split into a finite-dimensional covariate subspace (S)—representing the observable system—and its orthogonal complement (

S^{⊥}

). Through this decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as

G_{f}

, which acts as the computable “Hilbertian slice” of the otherwise intractable metric functional. Key theoretical contributions include proving the Trace Theorem (

H_{G} (f) = Tr (G_{f})

) to identify G-entropy as a fundamental geometric invariant; demonstrating the Geometric Invariance of the Covariate Fisher Information Matrix (cFIM) as a covariant

(0, 2)

-tensor under reparameterization; establishing the cFIM as the local Hessian of the KL-divergence; and characterizing the Efficiency Standard through a generalized Cramer–Rao Lower Bound for semi-parametric inference within the Orlicz manifold. Furthermore, we demonstrate that this framework provides a formal mathematical justification for the Manifold Hypothesis, as the structural decomposition naturally identifies the low-dimensional subspace where information is concentrated. By shifting the focus from the intractable global manifold to the tractable covariate geometry, this framework proves that statistical information is not a property of data alone, but an active geometric interaction between the environment (data), the system (covariate subspace), and the mechanism (Fisher–Rao connection). Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

12 pages, 570 KB

Open AccessArticle

Generalized Legendre Transforms Have Roots in Information Geometry

by Frank Nielsen

Entropy 2026, 28(1), 44; https://doi.org/10.3390/e28010044 - 30 Dec 2025

Viewed by 651

Abstract

Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661–674] characterized invertible reverse-ordering transforms in the space of lower, semi-continuous, extended, real-valued convex functions as affine deformations of the ordinary Legendre transform. In this work, we first prove that all those generalized Legendre transforms of functions correspond to the ordinary Legendre transform of dually corresponding affine-deformed functions. In short, generalized convex conjugates are ordinary convex conjugates of dually affine-deformed functions. Second, we explain how these generalized Legendre transforms can be derived from the dual Hessian structures of information geometry. Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

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26 pages, 360 KB

Open AccessFeature PaperArticle

Entropic Dynamics Approach to Quantum Electrodynamics

by Ariel Caticha

Entropy 2025, 27(12), 1247; https://doi.org/10.3390/e27121247 - 11 Dec 2025

Viewed by 544

Abstract

Entropic Dynamics (ED) is a framework that allows one to derive quantum theory as a Hamilton–Killing flow on the cotangent bundle of a statistical manifold. These flows are such that they preserve the symplectic and the metric geometries; they explain the linearity of quantum mechanics and the appearance of complex numbers. In this paper the ED framework is extended to deal with local gauge symmetries. More specifically, on the basis of maximum entropy methods and information geometry, for an appropriate choice of ontic variables and constraints, we derive the quantum dynamics of radiation fields interacting with charged particles. Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

25 pages, 17533 KB

Open AccessFeature PaperArticle

Mirror Descent and Exponentiated Gradient Algorithms Using Trace-Form Entropies

by Andrzej Cichocki, Toshihisa Tanaka, Frank Nielsen and Sergio Cruces

Entropy 2025, 27(12), 1243; https://doi.org/10.3390/e27121243 - 8 Dec 2025

Viewed by 1263

Abstract

This paper introduces a broad class of Mirror Descent (MD) and Generalized Exponentiated Gradient (GEG) algorithms derived from trace-form entropies defined via deformed logarithms. Leveraging these generalized entropies yields MD and GEG algorithms with improved convergence behavior, robustness against vanishing and exploding gradients, and inherent adaptability to non-Euclidean geometries through mirror maps. We establish deep connections between these methods and Amari’s natural gradient, revealing a unified geometric foundation for additive, multiplicative, and natural gradient updates. Focusing on the Tsallis, Kaniadakis, Sharma–Taneja–Mittal, and Kaniadakis–Lissia–Scarfone entropy families, we show that each entropy induces a distinct Riemannian metric on the parameter space, leading to GEG algorithms that preserve the natural statistical geometry. The tunable parameters of deformed logarithms enable adaptive geometric selection, providing enhanced robustness and convergence over classical Euclidean optimization. Overall, our framework unifies key first-order MD optimization methods under a single information-geometric perspective based on generalized Bregman divergences, where the choice of entropy determines the underlying metric and dual geometric structure. Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

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15 pages, 545 KB

Open AccessArticle

Geometry of Statistical Manifolds

by Paul W. Vos

Entropy 2025, 27(11), 1110; https://doi.org/10.3390/e27111110 - 27 Oct 2025

Viewed by 1305

Abstract

A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle [...] Read more.

T M

. Recognizing that points in M are distributions and not just generic points in a manifold,

T M

can be extended to a Hilbert bundle

H M

. This extension proves fundamental when we generalize the classical notion of a point estimate—a single point in M—to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization

θ : M \to Θ \subset R^{k}

\hat{θ}

is a distribution on

Θ

while its generalization

g_{\hat{θ}} = \hat{θ} - E \hat{θ}

as an estimate is a function over

Θ

that indicates inconsistency between the model and data. As an estimator,

g_{\hat{θ}}

is a distribution of functions. Geometric properties of these functions describe statistical properties of

g_{\hat{θ}}

. In particular, the expected slopes of

g_{\hat{θ}}

are used to define

Λ (g_{\hat{θ}})

, the

Λ

-information of

g_{\hat{θ}}

. The Fisher information I is an upper bound for the

Λ

-information: for all g,

Λ (g) \leq I

. We demonstrate the utility of this geometric perspective using the two-sample problem. Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

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