A note on the Artstein-Avidan-Milman's generalized Legendre transforms

TL;DR

This paper explores the generalized Legendre transforms introduced by Artstein-Avidan and Milman, showing they correspond to ordinary Legendre transforms on affine-deformed functions and linking them to dual Hessian structures in information geometry.

Contribution

It proves that all generalized Legendre transforms are equivalent to ordinary convex conjugates of dually affine-deformed functions and connects these transforms to dual Hessian structures in information geometry.

Findings

Generalized Legendre transforms correspond to ordinary convex conjugates of affine-deformed functions.

All such transforms are affine deformations of the standard Legendre transform.

Connection established between generalized transforms and dual Hessian structures in information geometry.

Abstract

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.