Symplectic Bregman divergences
This work provides a theoretical extension for researchers in geometric mechanics, information geometry, and machine learning, but it is incremental as it builds on existing Bregman divergence concepts.
The authors tackled the problem of generalizing Bregman divergences to symplectic vector spaces, resulting in the introduction of symplectic Bregman divergences derived from a symplectic Fenchel-Young inequality and symplectic subdifferentials.
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.