Divergence Functions and Geometric Structures They Induce on a Manifold

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, nonnegative function on the product manifold M ×M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold ∆M ⊂ M ×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 M; ii) a symplectic structure on M ×M if they are “proper”; iii) a Kähler structure on M×M if they further satisfy a certain condition. It is then shown that the class of DΦ-divergence functions (Zhang, 2004), as induced by a strictly convex function Φ on M, satisfies all these requirements and hence makes M×M a Kähler manifold (with Kähler potential given by Φ). This provides a larger context for the α-Hessian structure induced by the DΦ-divergence on M, which is shown to be equiaffine admitting α-parallel volume forms and biorthogonal coordinates generated by Φ and its convex conjugate Φ∗. As the α-Hessian structure is dually flat for α = ±1, the DΦ-divergence provides a richer geometric structures (compared to Bregman divergence) to the manifold M on which it is defined.