Divergence functions and their relation to metrical, equiaffine and symplectic structures on the manifold

  • Jun Zhang (University of Michigan/AFOSR, Ann Arbor, USA)
University n.n. Universität Leipzig (Leipzig)


Given a manifold $M$, a divergence function $D$ is a non-negative function on the product manifold $M \times M$ that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold $M_x$. It is well-known that the statistical structure on $M$ (a Riemmanian metric with a pair of conjugate affine connections) can be constructed from the second and third derivatives of $D$ evaluated at $M_x$. In Zhang (2004) and subsequent work, a framework based on convex analysis is proposed to unify familiar families of divergence functions. The resulting geometry, which displays what is called ``reference-representation biduality'', completely captures the alpha-structure (i.e., statistical structure with a parametric family of conjugate connections) of the classical information geometry. This is the alpha-Hessian geometry with equi-affine structure. Here, we continue this investigation in two parallel fronts, namely, how $D$ on $M \times M$ a) is related to various Minkowski metrics on $M$; and b) generates a symplectic structure of $M$. On point a, a set of inequalities are developed that uniformly bounds $D$ by Minkowski distances on $M$. On point b, convex-induced divergence functions will be shown to generate a K\"ahler structure under which the statistical structure of $M$ can be modeled.

02.08.10 06.08.10

Information Geometry and its Applications III

Universität Leipzig (Leipzig) University n.n. University n.n.

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Nihat Ay

Max Planck Institute for Mathematics in the Sciences, Germany

Paolo Gibilisco

Università degli Studi di Roma "Tor Vergata", Italy

František Matúš

Academy of Sciences of the Czech Republic, Czech Republic