Nielsen Frank
Sony Computer Science Laboratories Inc., Tokyo 141-0022, Japan.
Entropy (Basel). 2024 Dec 16;26(12):1101. doi: 10.3390/e26121101.
We present a generalization of Bregman divergences in finite-dimensional 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 built generically from pairings of dual systems, we obtain 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.
我们提出了有限维辛向量空间中布雷格曼散度的一种推广形式，我们称之为辛布雷格曼散度。辛布雷格曼散度源自芬切尔 - 杨不等式的辛推广，该推广依赖于辛次微分的概念。辛芬切尔 - 杨不等式是通过关于辛形式定义的辛芬切尔变换得到的。由于辛形式通常可以由对偶系统的配对构建而成，我们通过等价的辛布雷格曼散度得到了对偶系统中布雷格曼散度的一种推广。特别地，当辛形式由内积导出时，我们表明相应的辛布雷格曼散度相当于关于复合内积的普通布雷格曼散度。我们还提及了辛散度在几何力学、信息几何以及机器学习中的学习动力学方面的一些潜在应用。
