Symplectic Bregman divergences

TL;DR

This paper introduces symplectic Bregman divergences, a novel generalization of Bregman divergences in symplectic vector spaces, with potential applications in mechanics, geometry, and machine learning.

Contribution

It develops a symplectic generalization of the Fenchel-Young inequality and Bregman divergences, extending their applicability to dual systems and symplectic geometry.

Findings

Symplectic Bregman divergences reduce to classical Bregman divergences when derived from inner products.

The framework connects symplectic geometry with divergence measures in information geometry.

Potential applications include geometric mechanics and learning dynamics in machine learning.

Abstract

We present a generalization of Bregman divergences in symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel-Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel-Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be generically built from pairings of dual systems, we get a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon.