Generalized Legendre Transforms Have Roots in Information Geometry

Area of Science:

• Convex Analysis
• Information Geometry
• Functional Analysis

Background:

• The Legendre transform is a fundamental tool in convex analysis and its applications.
• Artstein-Avidan and Milman previously characterized specific invertible transforms as affine deformations of the Legendre transform.

Purpose of the Study:

• To establish a direct correspondence between generalized Legendre transforms and the ordinary Legendre transform.
• To demonstrate the derivation of these generalized transforms from information geometry's dual Hessian structures.

Main Methods:

• Proving the equivalence of generalized Legendre transforms to the ordinary Legendre transform of dually affine-deformed functions.
• Utilizing the dual Hessian structures inherent in information geometry.

Main Results:

• All generalized Legendre transforms studied correspond to the ordinary Legendre transform of dually affine-deformed functions.
• Generalized convex conjugates are shown to be ordinary convex conjugates of dually affine-deformed functions.
• A method for deriving these generalized transforms from information geometry is presented.

Conclusions:

• The findings provide a unified perspective on generalized Legendre transforms.
• This work connects convex analysis with information geometry through the lens of Legendre transforms.