揭示广义Legendre变换的本质：是变形函数的普通共轭
A note on the Artstein-Avidan-Milman's generalized Legendre transforms
- 将广义Legendre变换等价为变形函数的普通共轭
- 证明所有广义变换对应于对偶仿射变形函数的普通变换
- 连接信息几何中的双重Hessian结构，解释其数学来源
Artstein-Avidan与Milman（2009）刻画了下半连续实值凸函数空间上所有可逆反序变换均为普通Legendre变换的仿射变形。本文首先证明：所有这些广义Legendre变换在函数层面上，等价于对偶仿射变形函数上的普通凸共轭。简言之，广义凸共轭即对偶仿射变形函数的普通凸共轭。其次，我们阐明这些广义Legendre变换如何源自信息几何中双重Hessian结构的推导。
原文摘要 · Abstract (English)
Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661-674] characterized invertible reverse-ordering transforms on the space of lower semi-continuous extended real-valued convex functions as affine deformations of the ordinary Legendre transform. In this work, we first prove that all those generalized Legendre transforms on functions correspond to the ordinary Legendre transform on dually corresponding affine-deformed functions: In short, generalized convex conjugates are ordinary convex conjugates of dually affine-deformed functions. Second, we explain how these generalized Legendre transforms can be derived from the dual Hessian structures of information geometry.
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