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Symplectic Bregman Divergences.

Frank Nielsen1

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

Entropy (Basel, Switzerland)
|January 8, 2025
PubMed
Summary
This summary is machine-generated.

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.

Keywords:
Moreau proximationdual systemduality productgeometric mechanicsinner productsymplectic Fenchel transformsymplectic formsymplectic matrix groupsymplectic subdifferential

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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.