Symplectic and Kähler Structures on Statistical Manifolds Induced from Divergence Functions

Abstract

Divergence functions play a central role in information geometry. Given a manifold ${\mathcal M}$, a divergence function ${\mathcal D}$ is a smooth, non-negative function on the product manifold ${\mathcal M} \times{\mathcal M}$ that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold ${\mathcal M}_x$. It is well-known (Eguchi, 1982) that the statistical structure on ${\mathcal M}$ (a Riemmanian metric with a pair of conjugate affine connections) can be constructed from the second and third derivatives of ${\mathcal D}$ evaluated at ${\mathcal M}_x$. Here, we investigate Riemannian and symplectic structures on ${\mathcal M} \times{\mathcal M}$ as induced from ${\mathcal D}$. We derive a necessary condition about ${\mathcal D}$ for ${\mathcal M} \times{\mathcal M}$ to admit a complex representation and thus become a Kähler manifold. In particular, Kähler potential is shown to be globally defined for the class of Φ-divergence induced by a strictly convex function Φ (Zhang, 2004). In such case, we recover α-Hessian structure on the diagonal manifold ${\mathcal M}_x$, which is equiaffine and displays the so-called “reference-representation biduality.”