No. 950

Title:

Geometry on positive Definite Matrices and V-potential function.

Author(s):

Atsumi, Ohara (Department of System Science, Osaka-university);
Eguchi, Shinto (The Institute of Statistical Mathematics)

Key words:

Bias; Statistical manifold; Divergence; Negative constant curvature; Affine differential geometry;

Abstract:

    In this paper we investigate the geometry of the space of positive definite matrices induced by the class of convex functions called V-potentials from the viewpoints of information and affine geometry. We show the geometry is invariant for the unimodular group action and naturally induces a foliated structure. Each leaf is proved to be a statistical manifold with a negative constant curvature and possesses the special decomposition property of a pseudo-distance function called the divergence. As an application to statistics, we finally give the correspondence between the geometries of positive definite matrices and elliptical distributions.