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An Elementary Introduction to Information Geometry.

Frank Nielsen¹

¹Sony Computer Science Laboratories, Tokyo 141-0022, Japan.

Entropy (Basel, Switzerland)

|December 8, 2020

Summary

This summary is machine-generated.

This survey explores information geometry, detailing differential-geometric structures of information manifolds and their applications in information sciences. It introduces core concepts for understanding these geometric frameworks in data analysis.

Keywords:

Bayesian hypothesis testing Fisher–Rao distance Hessian manifolds affine connection conjugate connections curvature and flatness differential geometry dual metric-compatible parallel transport dually flat manifolds exponential family gauge freedom information manifold metric compatibility metric tensor mixed parameterization mixture clustering mixture family parameter divergence separable divergence statistical divergence statistical invariance statistical manifold α-embeddings

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Area of Science:

Information Geometry
Differential Geometry
Information Sciences

Background:

Information manifolds are central to statistical inference and machine learning.
Understanding their geometric properties is crucial for advanced data analysis.

Purpose of the Study:

To present the fundamental differential-geometric structures of information manifolds.
To state the core theorem of information geometry.
To showcase applications of information manifolds in information sciences.

Main Methods:

Concise introduction to essential differential geometry concepts.
Descriptive analysis of information manifold structures.
Illustrative examples of use cases.

Main Results:

Detailed description of differential-geometric structures.
Statement of the fundamental theorem of information geometry.
Demonstration of information manifold utility in various information science domains.

Conclusions:

Information geometry provides a powerful framework for understanding data.
The geometric structures and theorems offer insights into statistical models.
Applications highlight the practical relevance in information sciences.