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Lagrangian Submanifolds of Symplectic Structures Induced by Divergence Functions
1Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, 35121 Padova, Italy.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
Divergence functions in Information Geometry define geometric structures on probability manifolds. This study explores their use in describing Maximum Entropy principle solutions via Lagrangian submanifolds.
Area of Science:
- Information Geometry
- Differential Geometry
- Probability Theory
Background:
- Divergence functions are fundamental in Information Geometry, enabling Riemannian metrics and dual connections on probability manifolds.
- They also canonically induce a symplectic structure on the product of probability manifolds, a less explored area.
- Recent contributions highlight the significance of this symplectic structure.
Purpose of the Study:
- To explore applications of the symplectic structure derived from divergence functions.
- To investigate Lagrangian submanifolds within this symplectic structure.
- To demonstrate the utility of these submanifolds in characterizing Maximum Entropy principle solutions.
Main Methods:
- Utilizing divergence functions to define geometric structures on probability manifolds.
- Constructing the canonical symplectic structure on the square of a probability manifold.
- Analyzing Lagrangian submanifolds of the induced symplectic structure.
Main Results:
- The study identifies a connection between divergence functions and symplectic structures.
- Lagrangian submanifolds are shown to exist within this symplectic space.
- These submanifolds are demonstrated to effectively describe the solution manifold of the Maximum Entropy principle.
Conclusions:
- Divergence functions provide a powerful framework for understanding geometric structures in probability theory.
- The identified symplectic structure and its Lagrangian submanifolds offer novel insights into the Maximum Entropy principle.
- This research opens avenues for further exploration at the intersection of Information Geometry and statistical mechanics.
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