第28回統計的機械学習セミナー(2016年2月4日)
Time: Feburary 4 (Thu), 2015. 16:00-17:00
Place:Seminar Room 2
Speaker:
Jun Zhang (Department of Psychology and Department of Mathematics,University of Michigan-Ann Arbor)
Title:
Symplectic and (Para)-Kahler Structures on Statistical Manifolds
Abstract:
We study the interaction of a torsion-free affine connection nabla with
three objects on a manifold M: a pseudo-Riemannian metric g, a
skew-symmetric symplectic form omega, and a tangent-bundle isomorphism
L, two special cases being L= J (almost complex structure, J^2 = -id)
and L=K (almost para-complex structure, K^2 = id). It is well known
that Codazzi coupling of nabla with g gives rise to the statistical
structure. It is shown here that Codazzi coupling of nabla with any
two of the compatible triple (g, omega, L) will lead to its coupling
with the remainder, which further gives rise to a (para-)Kahler
structure on the manifold. We call this Codazzi-(para-)Kahler
structure, which is a natural generalization of special (para-)Kahler
geometry, without requiring nabla to be flat. In fact, we prove a
general result that g-conjugate, omega-conjugate, and L-gauge
transformations of nabla, along with the identity transform, form a
4-element Klein group. This leads a Codazzi-(para-)Kahler manifold to
admit a pair of torsion-free connections compatible with the (g, omega,
L). Finally, we give an example of Codazzi-(para-)K\"ahler manifold,
namely, the alpha-Hessian structure studied in information geometry.
(Joint work with Teng Fei, MIT).