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Information Geometry and Its Applications

  • Shun-ichi Amari

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

Table of contents

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

  3. Introduction to Dual Differential Geometry

    1. Front Matter
      Pages 107-107
    2. Shun-ichi Amari
      Pages 109-130
    3. Shun-ichi Amari
      Pages 131-161
  4. Information Geometry of Statistical Inference

    1. Front Matter
      Pages 163-163
    2. Shun-ichi Amari
      Pages 165-177
    3. Shun-ichi Amari
      Pages 179-189
    4. Shun-ichi Amari
      Pages 215-227
  5. Applications of Information Geometry

    1. Front Matter
      Pages 229-229
    2. Shun-ichi Amari
      Pages 231-278
    3. Shun-ichi Amari
      Pages 315-358
  6. Back Matter
    Pages 359-373

About this book

Introduction

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.

Keywords

Dual differential geometry Information geometry Machine learning Mathematical neuroscience Natural gradient learning Signal processing

Authors and affiliations

  • Shun-ichi Amari
    • 1
  1. 1.Brain Science InstituteRIKENWakoJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-4-431-55978-8
  • Copyright Information Springer Japan 2016
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-4-431-55977-1
  • Online ISBN 978-4-431-55978-8
  • Series Print ISSN 0066-5452
  • Series Online ISSN 2196-968X
  • Buy this book on publisher's site