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

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.

Figures (5)

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

Theorems & Definitions (17)

• proof
• Definition 1: Supporting hyperplane
• proof
• proof
• Theorem 1: Quadratic polarity as transformed convex body of the Legendre polarity
• proof
• Theorem 2: Quadratic polarity as Legendre polarity on deformed convex body
• proof
• Definition 2: Polar Fenchel-Young divergence
• proof