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Symplectic Bregman Divergences
1Sony Computer Science Laboratories Inc., Tokyo 141-0022, Japan.
Entropy (Basel, Switzerland)
|January 8, 2025
Summary
We introduce symplectic Bregman divergences, a novel generalization of Bregman divergences in symplectic geometry. This framework extends to dual systems and has potential applications in machine learning and geometric mechanics.
Area of Science:
- Mathematics
- Geometry
- Machine Learning
Background:
- Bregman divergences are a fundamental concept in convex analysis and information geometry.
- Generalizations are needed to extend their applicability to broader mathematical structures.
Purpose of the Study:
- To introduce and define symplectic Bregman divergences.
- To explore their theoretical underpinnings and connections to existing inequalities.
- To identify potential applications in various scientific fields.
Main Methods:
- Generalizing Bregman divergences within finite-dimensional symplectic vector spaces.
- Deriving the generalization from a symplectic Fenchel-Young inequality.
- Utilizing symplectic subdifferentials and the symplectic Fenchel transform.
- Connecting to dual systems and inner product structures.
Main Results:
- Definition of symplectic Bregman divergences.
- Establishment of a symplectic Fenchel-Young inequality.
- Demonstration of generalization across dual systems.
- Special case showing equivalence to Bregman divergences with composite inner products.
Conclusions:
- Symplectic Bregman divergences offer a powerful new tool for analyzing geometric and information-theoretic structures.
- The framework has broad applicability, including geometric mechanics, information geometry, and machine learning dynamics.
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