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

TL;DR

This paper explores quadratic polarity and polar Fenchel-Young divergences, revealing new dualities in information geometry through matrix manipulations of convex bodies and extending divergence concepts.

Contribution

It introduces a novel framework linking quadratic polarity functionals with deformed Legendre polarities and defines new polar divergences generalizing Fenchel-Young divergences.

Findings

Quadratic polarity functionals can be expressed via linear algebra on matrices.

Polar divergences generalize Fenchel-Young and Bregman divergences.

Total Bregman divergences are shown as a special case of polar Fenchel-Young divergences.

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.