Information Geometry and Its Applications

Front Cover
Springer, Feb 2, 2016 - Mathematics - 373 pages
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.
 

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Contents

1 Manifold Divergence and Dually Flat Structure
3
2 Exponential Families and Mixture Families of Probability Distributions
31
3 Invariant Geometry of Manifold of Probability Distributions
51
4 αGeometry Tsallis qEntropy and PositiveDefinite Matrices
70
Part II Introduction to Dual Differential Geometry
107
5 Elements of Differential Geometry
109
6 Dual Affine Connections and Dually Flat Manifold
131
Part III Information Geometry of Statistical Inference
162
Estimating Function and Semiparametric Statistical Model
191
10 Linear Systems and Time Series
214
Part IV Applications of Information Geometry
228
11 Machine Learning
229
12 Natural Gradient Learning and Its Dynamics in Singular Regions
279
13 Signal Processing and Optimization
315
References
359
Index
371

7 Asymptotic Theory of Statistical Inference
165
8 Estimation in the Presence of Hidden Variables
178

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