用二次极性统一解释信息几何中的对偶关系，揭示了散度的新结构。
Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity
- 通过二次极性将凸体变换与线性代数结合，实现高效计算
- 极性散度推广了Fenchel-Young散度，涵盖经典Bregman散度
- 提出对偶共形因子，重构参考对偶关系，适合几何学习研究者
极性是射影几何中基本的对偶关系，将点映射为超平面，更一般地将k维凸体映射为(n−1−k)维凸体。本文首先证明，由二次极性泛函诱导的通用极性可表示为变形的Legendre极性或变形凸体的Legendre极性，可通过作用于齐次坐标的(n+2)×(n+2)矩阵以线性代数方式高效操作。其次，基于Legendre极性定义极性散度，发现其广义化了Fenchel-Young散度或等价的Bregman散度。该极性研究深化了信息几何中核心参考对偶的理解。最后，证明总Bregman散度可视为总极性Fenchel-Young散度，并通过双极共形因子新展现参考对偶结构。
原文摘要 · Abstract (English)
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.
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