Polarity is a fundamental reciprocal duality of $n$-dimensional projective geometry which associates to points polar hyperplanes, and more generally $k$-dimensional convex bodies to polar $(n-1-k)$-dimensional convex bodies. It is well-known that the Legendre-Fenchel transformation of functions can be interpreted from the polarity viewpoint of their graphs using an extra dimension. In this paper, we first show that generic polarities induced by quadratic polarity functionals can be expressed either as deformed Legendre polarity or as the Legendre polarity of deformed convex bodies, and be efficiently manipulated using linear algebra on $(n+2)\times (n+2)$ matrices operating on homogeneous coordinates. Second, we define polar divergences using the Legendre polarity and show that they generalize the Fenchel-Young divergence or equivalent Bregman divergence. This polarity study brings new understanding of the core reference duality in information geometry. Last, we show that the total Bregman divergences can be considered as a total polar Fenchel-Young divergence from which we newly exhibit the reference duality using dual polar conformal factors.

翻译：极性作为$n$维射影几何的一种基本对偶互反关系，将点关联于极超平面，更一般地，将$k$维凸体关联于极$(n-1-k)$维凸体。众所周知，函数的Legendre-Fenchel变换可通过引入额外维度，从其图的极性视角进行解释。本文首先证明，由二次极性泛函诱导的一般极性既可表达为变形的Legendre极性，也可表达为变形凸体的Legendre极性，并可利用作用于齐次坐标的$(n+2)\times (n+2)$矩阵上的线性代数进行高效操作。其次，我们利用Legendre极性定义极散度，并证明其推广了Fenchel-Young散度或等价的Bregman散度。此项极性研究为信息几何中的核心参考对偶性带来了新的理解。最后，我们证明全Bregman散度可视为一种全极Fenchel-Young散度，并由此首次利用对极共形因子揭示了参考对偶性。