Symplectic Bregman divergences

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 a linear symplectic form. When the symplectic form is built 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.