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Information Geometry and Its Applications

  • Book
  • © 2016

Overview

Authors:
  1. Shun-ichi Amari

    1. Brain Science Institute, RIKEN, Wako, Japan

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  • Introduces information geometry intuitively to readers without knowledge of differential geometry
  • Includes hot topics of applications to machine learning, signal processing, neural networks, and optimization
  • Applies information geometry to statistical inference and time-series analysis
  • Includes supplementary material: sn.pub/extras

Part of the book series: Applied Mathematical Sciences (AMS, volume 194)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Geometry of Divergence Functions: Dually Flat Riemannian Structure

    1. Front Matter

      Pages 1-1
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    2. Manifold, Divergence and Dually Flat Structure

      • Shun-ichi Amari
      Pages 3-30

    3. Exponential Families and Mixture Families of Probability Distributions

      • Shun-ichi Amari
      Pages 31-49

    4. Invariant Geometry of Manifold of Probability Distributions

      • Shun-ichi Amari
      Pages 51-69

    5. \(\alpha \)-Geometry, Tsallis q-Entropy and Positive-Definite Matrices

      • Shun-ichi Amari
      Pages 71-106

  3. Introduction to Dual Differential Geometry

    1. Front Matter

      Pages 107-107
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    2. Elements of Differential Geometry

      • Shun-ichi Amari
      Pages 109-130

    3. Dual Affine Connections and Dually Flat Manifold

      • Shun-ichi Amari
      Pages 131-161

  4. Information Geometry of Statistical Inference

    1. Front Matter

      Pages 163-163
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    2. Asymptotic Theory of Statistical Inference

      • Shun-ichi Amari
      Pages 165-177

    3. Estimation in the Presence of Hidden Variables

      • Shun-ichi Amari
      Pages 179-189

    4. Neyman-Scott Problem: Estimating Function and Semiparametric Statistical Model

      • Shun-ichi Amari
      Pages 191-213

    5. Linear Systems and Time Series

      • Shun-ichi Amari
      Pages 215-227

  5. Applications of Information Geometry

    1. Front Matter

      Pages 229-229
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    2. Machine Learning

      • Shun-ichi Amari
      Pages 231-278

    3. Natural Gradient Learning and Its Dynamics in Singular Regions

      • Shun-ichi Amari
      Pages 279-314

    4. Signal Processing and Optimization

      • Shun-ichi Amari
      Pages 315-358

  6. Correction to: Information Geometry and Its Applications

    • Shun-ichi Amari
    Pages C1-C2

  7. Back Matter

    Pages 359-374
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Keywords

About this book

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.

Reviews

“This book gives a reasonably accessible introduction to the subject and then considers various applications. … The book provides a nice introduction to the subject. … the book provides a nice introduction to a difficult subject that has many important applications.” (Marvin H. J. Gruber, Technometrics, Vol. 58 (4), April, 2016) 

Authors and Affiliations

  • Brain Science Institute, RIKEN, Wako, Japan

    Shun-ichi Amari

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