cs.CGMar 5, 2026

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

Authors: Frank Nielsen, Basile Plus-Gourdon, Mahito Sugiyama

Organizations: Sony Computer Science Laboratories Inc. Tokyo, Japan · École Normale Supérieure Paris-Saclay, France · National Institute for Informatics Tokyo, Japan

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Abstract

Polarity is a fundamental reciprocal duality of nn-dimensional projective geometry which associates to points polar hyperplanes, and more generally kk-dimensional convex bodies to polar (n−1−k)(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)(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.

Figures & tables

Figure 1: A nn -variate function is represented by its (n+1)(n+1) -dimensional epigraph (convex body) and manipulated using homogeneous coordinates of Rn+2{\mathbb{R}}^{n+2} .
Figure 2: The graph of the Legendre transform F∗F^{*} of a function FF is the envelope of a family of polar hyperplanes called. Each hyperplane in blue is associated to a point (θ,F(θ))∈graph(F)(\theta,F(\theta))\in{\mathrm{graph}(F)} . Here, we consider F(θ)=θ2+θ+3F(\theta)=\theta^{2}+\theta+3 with convex conjugate F∗(η)=14(η2−2η−13)F^{*}(\eta)=\frac{1}{4}(\eta^{2}-2\eta-13) .
Figure 3: Illustration of the quadratic polarity ΔC:Rn+1→Rn+1\Delta_{C}:\mathbb{R}^{n+1}\to\mathbb{R}_{n+1} induced by (n+2)×(n+2)(n+2)\times(n+2) matrix CC : Polarity maps a set A⊂Rn+1A\subset\mathbb{R}^{n+1} to its dual ΔC(A)⊂Rn+1\Delta_{C}(A)\subset\mathbb{R}_{n+1} . A point [b][b] in Rn+1\mathbb{R}_{n+1} can be represented in Rn+1\mathbb{R}^{n+1} as a hyperplane H[b]H_{[b]} of normal vector C [b]C\,[b] .
Figure 4: Ordinary Fenchel-Young divergence from the viewpoint of Legendre polarity: [a]∈∂A[a]\in\partial A and [b]∈∂Δ(A)[b]\in\partial\Delta(A) .
Figure 5: Generalized Fenchel-Young divergence: [a]∈A[a]\in A and [b]∈Δ(A)[b]\in\Delta(A)
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