Methods of Information Geometry

2000
Methods of Information Geometry

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  • Author: Shun-ichi Amari
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821843024
  • Category : Computers
  • Languages : en
  • Pages : 220

Information geometry provides the mathematical sciences with a fresh framework of analysis. This book presents a comprehensive introduction to the mathematical foundation of information geometry. It provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, and convex analysis.

Information Geometry and Its Applications

2016-02-02
Information Geometry and Its Applications

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  • Author: Shun-ichi Amari
  • Publisher: Springer
  • ISBN: 4431559787
  • Category : Mathematics
  • Languages : en
  • Pages : 378

This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.

Computational Information Geometry

2016-11-24
Computational Information Geometry

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  • Author: Frank Nielsen
  • Publisher: Springer
  • ISBN: 3319470582
  • Category : Technology & Engineering
  • Languages : en
  • Pages : 299

This book focuses on the application and development of information geometric methods in the analysis, classification and retrieval of images and signals. It provides introductory chapters to help those new to information geometry and applies the theory to several applications. This area has developed rapidly over recent years, propelled by the major theoretical developments in information geometry, efficient data and image acquisition and the desire to process and interpret large databases of digital information. The book addresses both the transfer of methodology to practitioners involved in database analysis and in its efficient computational implementation.

Information Geometry

2019-04-04
Information Geometry

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  • Author: Geert Verdoolaege
  • Publisher: MDPI
  • ISBN: 3038976326
  • Category : Juvenile Nonfiction
  • Languages : en
  • Pages : 355

This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience.

Information Geometry

2017-08-25
Information Geometry

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  • Author: Nihat Ay
  • Publisher: Springer
  • ISBN: 3319564781
  • Category : Mathematics
  • Languages : en
  • Pages : 407

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated. This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems.

Information Geometry

2021
Information Geometry

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  • Author:
  • Publisher: Springer Science & Business Media
  • ISBN: 3540693912
  • Category :
  • Languages : en
  • Pages : 263

Methods of Geometry

2011-03-01
Methods of Geometry

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  • Author: James T. Smith
  • Publisher: John Wiley & Sons
  • ISBN: 1118031032
  • Category : Mathematics
  • Languages : en
  • Pages : 486

A practical, accessible introduction to advanced geometryExceptionally well-written and filled with historical andbibliographic notes, Methods of Geometry presents a practical andproof-oriented approach. The author develops a wide range ofsubject areas at an intermediate level and explains how theoriesthat underlie many fields of advanced mathematics ultimately leadto applications in science and engineering. Foundations, basicEuclidean geometry, and transformations are discussed in detail andapplied to study advanced plane geometry, polyhedra, isometries,similarities, and symmetry. An excellent introduction to advancedconcepts as well as a reference to techniques for use inindependent study and research, Methods of Geometry alsofeatures: Ample exercises designed to promote effective problem-solvingstrategies Insight into novel uses of Euclidean geometry More than 300 figures accompanying definitions and proofs A comprehensive and annotated bibliography Appendices reviewing vector and matrix algebra, least upperbound principle, and equivalence relations An Instructor's Manual presenting detailed solutions to all theproblems in the book is available upon request from the Wileyeditorial department.

Differential-Geometrical Methods in Statistics

2012-12-06
Differential-Geometrical Methods in Statistics

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  • Author: Shun-ichi Amari
  • Publisher: Springer Science & Business Media
  • ISBN: 1461250560
  • Category : Mathematics
  • Languages : en
  • Pages : 302

From the reviews: "In this Lecture Note volume the author describes his differential-geometric approach to parametrical statistical problems summarizing the results he had published in a series of papers in the last five years. The author provides a geometric framework for a special class of test and estimation procedures for curved exponential families. ... ... The material and ideas presented in this volume are important and it is recommended to everybody interested in the connection between statistics and geometry ..." #Metrika#1 "More than hundred references are given showing the growing interest in differential geometry with respect to statistics. The book can only strongly be recommended to a geodesist since it offers many new insights into statistics on a familiar ground." #Manuscripta Geodaetica#2

Algebraic and Geometric Methods in Statistics

2010
Algebraic and Geometric Methods in Statistics

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  • Author: Paolo Gibilisco
  • Publisher: Cambridge University Press
  • ISBN: 0521896193
  • Category : Mathematics
  • Languages : en
  • Pages : 447

An up-to-date account of algebraic statistics and information geometry, which also explores the emerging connections between these two disciplines.

Geometric Structures of Statistical Physics, Information Geometry, and Learning

2021-06-27
Geometric Structures of Statistical Physics, Information Geometry, and Learning

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  • Author: Frédéric Barbaresco
  • Publisher: Springer Nature
  • ISBN: 3030779572
  • Category : Mathematics
  • Languages : en
  • Pages : 466

Machine learning and artificial intelligence increasingly use methodological tools rooted in statistical physics. Conversely, limitations and pitfalls encountered in AI question the very foundations of statistical physics. This interplay between AI and statistical physics has been attested since the birth of AI, and principles underpinning statistical physics can shed new light on the conceptual basis of AI. During the last fifty years, statistical physics has been investigated through new geometric structures allowing covariant formalization of the thermodynamics. Inference methods in machine learning have begun to adapt these new geometric structures to process data in more abstract representation spaces. This volume collects selected contributions on the interplay of statistical physics and artificial intelligence. The aim is to provide a constructive dialogue around a common foundation to allow the establishment of new principles and laws governing these two disciplines in a unified manner. The contributions were presented at the workshop on the Joint Structures and Common Foundation of Statistical Physics, Information Geometry and Inference for Learning which was held in Les Houches in July 2020. The various theoretical approaches are discussed in the context of potential applications in cognitive systems, machine learning, signal processing.