一般化ルジャンドル変換の情報幾何学的起源

Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661-674] characterized invertible reverse-ordering transforms in 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 of functions correspond to the ordinary Legendre transform of 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.