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

arXiv: https://arxiv.org/abs/2603.04812 · HTML: https://mlsanity.com/paper/2603.04812/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 28 references that are in the map.
- First posted on arXiv 5 Mar 2026.

## Builds on (most cited first, 4 of 4)

- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Hopfield-Fenchel-Young Networks: A Unified Framework for Associative Memory Retrieval](https://mlsanity.com/paper/2411.08590/) (2024)
- [Topological Data Analysis with Bregman Divergences](https://mlsanity.com/paper/1607.06274/) (2016)
- [A provable control of sensitivity of neural networks through a direct parameterization of the overall bi-Lipschitzness](https://mlsanity.com/paper/2404.09821/) (2024)

## Built on by (most cited first, 1 of 1)

- [Neural Legendre-Fenchel transform with Hessian Preconditioning](https://mlsanity.com/paper/2606.09077/) (2026)

## Abstract (arXiv)

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)× (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.