第28回統計的機械学習セミナー / The 28th Statistical Machine Learning Seminar

Date&Time
2016年2月4日(木)16:00-17:00
/ 4 February, 2016 (Thu) 16:00-17:00

Admission Free,No Booking Necessary

Place
統計数理研究所 セミナー室2 (D304)
/ Seminar Room 2 (D304) @ The Institute of Statistical Mathematics
区切り線
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).