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

Frank Nielsen

TL;DR

The paper shows that Artstein-Avidan and Milman’s generalized Legendre transforms can be realized as ordinary Legendre transforms applied to affine-deformed functions. It introduces a diamond operator that maps deformation parameters to an involutive pair, establishing that GLFTs are equivalent to LFTs on affine-deformed inputs and outputs. An information-geometric interpretation is developed: GLFTs arise naturally in dually flat Hessian manifolds, with dual potentials, Fenchel-Young divergences, and an equivalence relation on deformation classes clarifying the geometric origin of the extra degrees of freedom. This unifies generalized Legendre transforms with standard convex duality and situates them firmly within Hessian geometry and 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.

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

TL;DR

The paper shows that Artstein-Avidan and Milman’s generalized Legendre transforms can be realized as ordinary Legendre transforms applied to affine-deformed functions. It introduces a diamond operator that maps deformation parameters to an involutive pair, establishing that GLFTs are equivalent to LFTs on affine-deformed inputs and outputs. An information-geometric interpretation is developed: GLFTs arise naturally in dually flat Hessian manifolds, with dual potentials, Fenchel-Young divergences, and an equivalence relation on deformation classes clarifying the geometric origin of the extra degrees of freedom. This unifies generalized Legendre transforms with standard convex duality and situates them firmly within Hessian geometry and 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.

Paper Structure

This paper contains 11 sections, 4 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalized Legendre transforms as ordinary Legendre transforms
  3. Proof:
  4. Proof:
  5. Proof:
  6. Proof:
  7. An information-geometric interpretation of generalized Legendre transforms
  8. Conclusion
  9. Legendre-type functions
  10. Proof:
  11. Some examples of convex conjugates

Key Result

Theorem 1

Let $\mathcal{T}$ be an invertible transform such that Then $\mathcal{T}$ is a generalized Legendre-Fenchel transform (GLFT), written canonically as: where $\lambda>0$, $E\in\mathrm{GL}(\mathbb{R}^m)$ (general linear group), $f,g\in\mathbb{R}^m$ and $h\in\mathbb{R}$.

Figures (2)

  • Figure 1: The ordinary Legendre transform on various classes of functions: Relationships with Fenchel-Young and Bregman divergences, dually flat Hessian divergence, and $\alpha$-geometry in information geometry.
  • Figure 2: A pair $(F(\theta),F^*(\eta))$ of conjugate functions (top) with their subgradients plotted (bottom). Function $F(\theta)$ is not differentiable at $\theta=0$ and thus admits a subgradient $\partial F(0)$ at $\theta=0$. Function $F^*(\eta)$ is everywhere differentiable, and when $\theta\not=0$, we have $\nabla F^*=(\nabla F)^{-1}$ visualized by rotating the $xy$-axis 90-degrees).

Theorems & Definitions (11)

  • Theorem 1: AxiomatizationLegendreTransform-2009, Theorem 7
  • Proposition 1: LFT of an affine-deformed function
  • Proposition 2: $\diamond$-involution
  • Definition 1: Generalized Legendre-Fenchel convex conjugates AxiomatizationLegendreTransform-2009
  • Theorem 2
  • Definition 2: Legendre-type function LegendreType-1967
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • ...and 1 more