Progress in information geometry : theory and applications / Frank Nielsen, editor.

Format:

Book

Contributor:

Nielsen, Frank, editor.

Series:

Signals and communication technology

Language:

English

Subjects (All):

Manifolds (Mathematics).

Genre:

Electronic books.

Physical Description:

1 online resource (282 pages).

Place of Publication:

Cham : Springer, [2021]

System Details:

text file

Summary:

This book focuses on information-geometric manifolds of structured data and models and related applied mathematics. It features new and fruitful interactions between several branches of science: Advanced Signal/Image/Video Processing, Complex Data Modeling and Analysis, Statistics on Manifolds, Topology/Machine/Deep Learning and Artificial Intelligence. The selection of applications makes the book a substantial information source, not only for academic scientist but it is also highly relevant for industry. The book project was initiated following discussions at the international conference GSI2019 Geometric Science of Information that was held at ENAC, Toulouse (France).

Contents:

Intro

Preface

Contents

1 Information Geometry of Smooth Densities on the Gaussian Space: Poincaré Inequalities

1.1 Introduction

1.2 Statistical Bundle Modelled on Orlicz Spaces

1.2.1 Orlicz Spaces

1.2.2 Calculus of the Gaussian Space

1.2.3 Exponential Statistical Bundle

1.3 Bounding the Orlicz Norm with the Orlicz Norm of the Gradient

1.3.1 Ornstein-Uhlenbeck Semi-group

1.3.2 Generator of the Ornstein-Uhlenbeck Semi-group

1.4 Discussion and Conclusions

1.4.1 Sub-exponential Random Variables

1.4.2 Hyvärinen Divergence

1.4.3 Otto's Metric

3 Affine Connections with Torsion in (Para-)complexified Structures

3.1 Introduction

3.2 Torsion of and Integrability of L

3.2.1 L Conjugation of

3.2.2 Nijenhuis Tensor NL and Integrability

3.2.3 MC1 Versus MC2

3.3 Torsion of Under (Para-)complexification

3.3.1 Splitting of TmathcalMotimesmathbbL by L

3.3.2 (Para-)holomorphicity of

3.3.3 (Para-)complexifying NL and T

3.3.4 Torsion-Compatibility (MC1)

3.3.5 Torsion-Coupling (MC2)

3.4 Summary and Discussions

References

4 Contact Hamiltonian Systems for Probability Distribution Functions and Expectation Variables: A Study Based on a Class of Master Equations

4.1 Introduction

4.2 Underlying Geometry

4.2.1 Para-Contact Metric Manifolds

4.2.2 Contact Manifold

4.2.3 Legendre Submanifold as Dually Flat Space

4.2.4 Legendre Submanifold as Attractor

4.3 Distribution Functions from Solvable Master Equations

4.3.1 Denormalization

4.4 Observables with Solvable Master Equations

4.5 Geometric Description of Master Equations

4.5.1 Geometry of Equilibrium States

4.5.2 Geometry of Nonequilibrium States

4.6 Geometric Description of Expectation Variables

4.6.1 Geometry of Equilibrium States

4.6.2 Geometry of Nonequilibrium States

4.7 Beyond the Toy Model

4.7.1 Equilibrium States

4.7.2 Nonequilibrium States

4.8 Conclusions

References

5 Invariant Koszul Form of Homogeneous Bounded Domains and Information Geometry Structures

5.1 Preamble

5.2 Invariant Koszul Form for Homogeneous Bounded Domains

5.3 Contextualization with Last Advanced Works

5.4 Koszul Hessian Geometric Structure of Information Geometry

1.4.4 Conclusion and Acknowledgments

2 On Normalization Functions and -Families of Probability Distributions

2.1 Introduction

2.2 Revisiting Deformed Exponentials

2.2.1 Deformed Exponential Functions

2.2.2 -Families of Probability Distributions

2.2.3 The Δ2- Condition and -Families of Probability Distributions

2.3 The Behavior of ψ Near the Boundary of mathcalBc

2.3.1 Condition 2.2 is Satisfied

2.3.2 Condition 2.2 is Not Satisfied

2.3.3 Condition 2.2 is Not Satisfied: Behavior of the Normalizing Function

2.4 Conclusions

Notes:

Description based upon print version of record.

Other Format:

Print version: Nielsen, Frank Progress in Information Geometry

ISBN:

9783030654597

3030654591

OCLC:

1243544872

Access Restriction:

Restricted for use by site license.