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

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 note, we prove that all those generalized Legendre transforms on functions correspond to the ordinary Legendre transform on dually corresponding affine-deformed functions. That is, generalized convex conjugates are convex conjugates of affine-deformed functions. We conclude this note by sketching how this result can be interpreted from the lens of information geometry.

• The paper demonstrates that every generalized Legendre transform is equivalent to an ordinary Legendre-Fenchel transform applied to an affine-deformed convex function.
• It provides explicit parameterizations of affine deformations and establishes an involutive structure that preserves convexity and duality properties.
• The study interprets these findings within information geometry, linking dual coordinate systems with the invariance of divergence measures.

Generalized Legendre Transforms as Affine-Deformed Convex Conjugates

Introduction

This note provides a rigorous analysis of the Artstein-Avidan-Milman (AAM) characterization of generalized Legendre transforms (GLFTs) on the space of proper lower semi-continuous convex functions. The main result establishes that all such GLFTs are, in fact, ordinary Legendre-Fenchel transforms (LFTs) applied to affine-deformed functions. This equivalence is formalized through explicit parameterizations and involutive transformations, and the implications are further interpreted within the framework of information geometry, particularly in the context of dually flat spaces and their associated divergences.

Theoretical Foundations

Let $\Gamma_0$ denote the space of proper, lower semi-continuous, extended real-valued convex functions on $\mathbb{R}^m$. The Legendre-Fenchel transform $L F$ of $F \in \Gamma_0$ is defined as

$$(L F)(\eta) = \sup_{\theta \in \mathbb{R}^m} \left\{ \langle \theta, \eta \rangle - F(\theta) \right\}.$$

The biconjugate property $(F^*)^* = F$ holds for $F \in \Gamma_0$, and the transform is order-reversing.

AAM's theorem characterizes all invertible, order-reversing transforms $T$ on $\Gamma_0$ as affine deformations of the LFT: $$(T F)(\eta) = \lambda (L F)(E\eta + f) + \langle \eta, g \rangle + h,$$ where $\lambda > 0$, $E \in GL(\mathbb{R}^m)$, $f, g \in \mathbb{R}^m$, and $h \in \mathbb{R}$.

Affine Deformations and Convexity Preservation

Affine deformations of a function $F$ are parameterized as

$$F_P(\theta) = \lambda F(A\theta + b) + \langle \theta, c \rangle + d,$$

with $P = (\lambda, A, b, c, d)$. Such deformations preserve convexity and lower semi-continuity, ensuring $F_P \in \Gamma_0$ for all admissible $P$.

Legendre Transform of Affine-Deformed Functions

The Legendre transform of $F_P$ yields another affine-deformed convex conjugate: $L(F_P) = (L F)_{P^\diamond},$ where the involutive parameter transformation $P \mapsto P^\diamond$ is given by

$$P^\diamond = \left( \lambda, \frac{1}{\lambda}A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b, A^{-1}c \rangle - d \right).$$

This involution property ensures that applying the transformation twice returns the original parameters, i.e., $(P^\diamond)^\diamond = P$.

Equivalence of Generalized and Ordinary Legendre Transforms

The central result is that any GLFT can be realized as an ordinary LFT on an affine-deformed function: $(T F)(\eta) = L(F_{P^\diamond})(\eta).$ This demonstrates that the apparent generality of GLFTs is subsumed by the structure of the ordinary LFT, provided one allows for affine changes of variables and scaling.

Figure 1: The ordinary Legendre transform on classes of functions: Relationships with representational Fenchel-Young and Bregman divergences, flat Hessian divergence, and $\alpha$-geometry in information geometry.

Subgradients and Reciprocal Gradients

The analysis extends to non-differentiable convex functions, where subgradients replace gradients. For Legendre-type functions (strictly convex, differentiable, and steep at the boundary), the gradients of conjugate pairs are reciprocal: $$\nabla F^* = (\nabla F)^{-1}, \quad \nabla F = (\nabla F^*)^{-1}.$$ This property is crucial for the geometric interpretation and for applications in optimization and information geometry.

Figure 2: A pair $(F(\theta),F^*(\eta))$ of conjugate functions (top) with their subgradients plotted (bottom). $F(\theta)$ is not differentiable at $\theta=0$ and thus admits a subgradient $\partial F(0)$ at $\theta=0$. $F^*(\eta)$ is everywhere differentiable, and when $\theta \neq 0$, $\nabla F^* = (\nabla F)^{-1}$.

Information-Geometric Interpretation

The result is interpreted through the lens of information geometry, where a strictly convex function $F$ induces a dually flat manifold $(M, g, \nabla, \nabla^*)$. The affine freedom in the choice of coordinates and potential functions corresponds precisely to the affine deformations in the GLFT characterization. The Fenchel-Young inequality and the associated divergences (Fenchel-Young, Bregman, and dually flat divergences) are invariant under these affine transformations, leading to an equivalence relation on the moduli space of dually flat spaces.

The parameter $\lambda$ corresponds to scaling the metric and connections, reflecting the geometric invariance of the divergence structure under such rescalings. The duality between the primal and dual coordinate systems, and the associated potential functions, is preserved under the affine-deformed LFT framework.

Implications and Future Directions

The identification of all GLFTs as ordinary LFTs on affine-deformed functions has several implications:

• Convex Analysis: The result provides a complete classification of order-reversing involutive transforms on $\Gamma_0$, reducing the study of GLFTs to the well-understood theory of LFTs and affine transformations.
• Optimization: Algorithms that rely on convex conjugacy (e.g., in Fenchel duality, mirror descent, or variational inference) can be generalized to accommodate affine-deformed settings without loss of generality.
• Information Geometry: The affine invariance elucidated here underpins the geometric structure of dually flat spaces, with direct consequences for the study of divergences, exponential families, and statistical manifolds.
• Theoretical Generalization: The involutive structure and parameterization may inform further generalizations, such as to infinite-dimensional settings or to other classes of duality transforms.

Conclusion

This note rigorously demonstrates that the Artstein-Avidan-Milman generalized Legendre transforms are, in essence, ordinary Legendre-Fenchel transforms applied to affine-deformed convex functions. The involutive parameterization and the information-geometric interpretation provide a unified perspective that bridges convex analysis, optimization, and information geometry. This equivalence simplifies the theoretical landscape and offers a robust foundation for further developments in the analysis and application of convex duality and geometric structures.