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
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.
Figures & tables
Figure 1: A n -variate function is represented by its (n+1) -dimensional epigraph (convex body) and manipulated using homogeneous coordinates of Rn+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 (θ,F(θ))∈graph(F) . Here, we consider F(θ)=θ2+θ+3 with convex conjugate F∗(η)=41(η2−2η−13) .
Figure 3: Illustration of the quadratic polarity ΔC:Rn+1→Rn+1 induced by (n+2)×(n+2) matrix C : Polarity maps a set A⊂Rn+1 to its dual ΔC(A)⊂Rn+1 . A point [b] in Rn+1 can be represented in Rn+1 as a hyperplane H[b] of normal vector C[b] .
Figure 4: Ordinary Fenchel-Young divergence from the viewpoint of Legendre polarity: [a]∈∂A and [b]∈∂Δ(A) .
Figure 5: Generalized Fenchel-Young divergence: [a]∈A and [b]∈Δ(A)
This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.
Tushar Sial, Abhishek Halder
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA
We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.
Symmetric positive-definite (SPD) matrix datasets play a central role across numerous scientific disciplines, including signal processing, statistics, finance, computer vision, information theory, and machine learning among others. The set of SPD matrices forms a cone which can be viewed as a global coordinate chart of the underlying SPD manifold. Rich differential-geometric structures may be defined on the SPD cone manifold. Among the most widely used geometric frameworks on this manifold are the affine-invariant Riemannian structure and the dual information-geometric log-determinant barrier structure, each associated with dissimilarity measures (distance and divergence, respectively). In this work, we introduce two new structures, a Finslerian structure and a dual information-geometric structure, both derived from James' bicone reparameterization of the SPD domain. Those structures ensure that geodesics correspond to straight lines in appropriate coordinate systems. The closed bicone domain includes the spectraplex (the set of positive semi-definite diagonal matrices with unit trace) as an affine subspace, and the Hilbert VPM distance is proven to generalize the Hilbert simplex distance which found many applications in machine learning. Finally, we discuss several applications of these Finsler/dual Hessian structures and provide various inequalities between the new and traditional dissimilarities.
Jacek Karwowski, Frank Nielsen
Department of Computer Science University of Oxford, UK · Sony Computer Science Laboratories Inc. Tokyo, Japan