Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity

2026-03-05

AI Summary

This work proposes a unified perspective on the Legendre–Fenchel transform, Fenchel–Young divergence, and their dual structures in information geometry through the lens of polarity in projective geometry. By introducing a quadratic polarity functional, general polarity is expressed as a deformed Legendre polarity, enabling efficient computation via $(n+2)\times(n+2)$ homogeneous coordinate matrices. Building on this framework, the authors define a polarity divergence that generalizes both Fenchel–Young and Bregman divergences. Notably, they reveal for the first time that the total Bregman divergence can be interpreted as a total polarity Fenchel–Young divergence, with its reference duality clarified through a polarity conformal factor. This approach establishes a cohesive framework for polarity-based divergences, encompassing classical instances while offering new insights into duality theory in information geometry.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

Polarity is a fundamental reciprocal duality of $n$-dimensional projective geometry which associates to points polar hyperplanes, and more generally $k$-dimensional convex bodies to polar $(n-1-k)$-dimensional convex bodies. It is well-known that the Legendre-Fenchel transformation of functions can be interpreted from the polarity viewpoint of their graphs using an extra dimension. In this paper, we first show that generic polarities induced by quadratic polarity functionals can be expressed either as deformed Legendre polarity or as the Legendre polarity of deformed convex bodies, and be efficiently manipulated using linear algebra on $(n+2)\times (n+2)$ matrices operating on homogeneous coordinates. Second, we define polar divergences using the Legendre polarity and show that they generalize the Fenchel-Young divergence or equivalent Bregman divergence. This polarity study brings new understanding of the core reference duality in information geometry. Last, we show that the total Bregman divergences can be considered as a total polar Fenchel-Young divergence from which we newly exhibit the reference duality using dual polar conformal factors.

Problem

• polarity
• Legendre-Fenchel transformation
• Fenchel-Young divergence
• Bregman divergence
• information geometry

Innovation

• quadratic polarity
• Legendre polarity
• polar Fenchel-Young divergence
• total Bregman divergence
• information geometry