Divergence Functions and Geometric Structures They Induce on a Manifold

Abstract

Divergence functions play a central role in information geometry. Given a manifold $\mathfrak {M}$, a divergence function$\mathcal {D}$ is a smooth, nonnegative function on the product manifold $\mathfrak {M}\times \mathfrak {M}$ that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold $\varDelta _{\mathfrak {M}} \subset \mathfrak {M}\times \mathfrak {M}$. In this chapter, we review how such divergence functions induce (i) a statistical structure (i.e., a Riemannian metric with a pair of conjugate affine connections) on $\mathfrak {M}$; (ii) a symplectic structure on $\mathfrak {M}\times \mathfrak {M}$ if they are “proper”; (iii) a Kähler structure on $\mathfrak {M}\times \mathfrak {M}$ if they further satisfy a certain condition. It is then shown that the class of $\mathcal {D}_\varPhi$-divergence functions [23], as induced by a strictly convex function$\varPhi$ on $\mathfrak {M}$, satisfies all these requirements and hence makes $\mathfrak {M}\times \mathfrak {M}$ a Kähler manifold (with Kähler potential given by $\varPhi$). This provides a larger context for the $\alpha$-Hessian structure induced by the $\mathcal {D}_\varPhi$-divergence on $\mathfrak {M}$, which is shown to be equiaffine admitting $\alpha$-parallel volume forms and biorthogonal coordinates generated by $\varPhi$ and its convex conjugate $\varPhi ^{*}$. As the $\alpha$-Hessian structure is dually flat for $\alpha = \pm 1$, the $\mathcal {D}_\varPhi$-divergence provides richer geometric structures (compared to Bregman divergence) to the manifold $\mathfrak {M}$ on which it is defined.