Jihun Yum, Stochastic Bergman Geometry

For a bounded domain Ω in Cⁿ, let P(Ω) be the set of all (real) probability distributions on Ω. Then, in general, P(Ω) becomes an infinite-dimensional smooth manifold and it always admit a natural Riemannian pseudo-metric, called the Fisher information metric, on P(Ω). Information geometry studies a finite-dimensional submanifold M, which is called a statistical model, in P(Ω) using geometric concepts such as Riemannian metric, distance, connection, and curvature, to better understand the properties of statistical models M and provide insights into the behavior of learning algorithms and optimization methods.

In this talk, we first introduce a map Φ : Ω → P(Ω) and prove that the pull-back of the Fisher information metric on P(Ω) is exactly same as the Bergman metric of Ω. This map provides a completely new perspective that allows us to view Bergman geometry from a stochastical viewpoint. We will discuss the following 4 things.

1. The relation between Φ and the Kobayashi map ι : Ω → CP^∞.

2. A Stochastic formula for the holomorphic sectional curvature of the Bergman metric.

3. A Stochastic condition for injectivity of a proper holomorphic surjective map between two bounded domains.

4. The central limit theorem on Ω.

This is a joint work with Gunhee Cho at UC Santa Barbara University.