# Generalized Legendre Transforms Have Roots in Information Geometry

## TL;DR

This paper explores how generalized Legendre transforms relate to information geometry and are derived from affine deformations of standard transforms.

## Contribution

The paper shows that generalized Legendre transforms are ordinary transforms of dually affine-deformed functions and connects them to information geometry.

## Key findings

- Generalized Legendre transforms correspond to ordinary transforms of dually affine-deformed functions.
- These transforms are derived from dual Hessian structures in information geometry.

## Abstract

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.