Prof. Jun Zhang
(University of Michigan-Ann Arbor)
Title
Some Recent Progress of Information Geometry
Abstract
Classical information geometry constructs a pair (and hence an alpha-family) of torsion-free affine-connections that are conjugate with respect to the Fisher-Rao metric.
And when the conjugate connections are furthermore curvature-free, Hessian manifolds result for the exponential family of probability models.
This talk reports recent advances in extending this classical framework.
First, we generalize the notion of biorthogonal coordinates (of a Hessian manifold) to biorthogonal frames for any statistical manifold.
The pair of connections adapted to biorthogonal frames turn out to be conjugate to each other and, in general, they are curvature-free but carry torsions.
We then construct a generic “partially-flat” geometry of parametric probability model in which one of the connections is flat and the other curvature-free but torsion-carrying.
This partially-flat SMAT (statistical manifold admitting torsion) geometry, a la Henmi and Matsuzoe,
becomes Hessian (dually flat) when the canonically-constructed frame (biorthogonal to the natural coordinates) becomes a coordinate frame.
Finally, we study compatibility of conjugate connections of statistical manifolds to almost (para-)complex structures,
and show how statistical structures may be enhanced to (para-) Kahler manifold and (para-)Hermitian manifolds.
(Above joint work with Teng Fei, Sergey Grigorian, Gabriel Khan). If time permits,
I will also report (joint work with Jan Naudts) on using rho-tau monotone embedding framework to provide a unifying framework of deformed exponential family of parametric probability manifold and the two geometries they generate (each under a different “gauge”).
