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(2014) Geometric theory of information, Dordrecht, Springer.
Geometry on positive definite matrices deformed by v-potentials and its submanifold structure
Atsumi Ohara, Shinto Eguchi
pp. 31-55
In this paper we investigate dually flat structure of the space of positive definite matrices induced by a class of convex functions called V-potentials, from a viewpoint of information geometry. It is proved that the geometry is invariant under special linear group actions and naturally introduces a foliated structure. Each leaf is proved to be a homogeneous statistical manifold with a negative constant curvature and enjoy a special decomposition property of canonically defined divergence. As an application to statistics, we finally give the correspondence between the obtained geometry on the space and the one on elliptical distributions induced from a certain Bregman divergence.
Publication details
DOI: 10.1007/978-3-319-05317-2_2
Full citation:
Ohara, A. , Eguchi, S. (2014)., Geometry on positive definite matrices deformed by v-potentials and its submanifold structure, in F. Nielsen (ed.), Geometric theory of information, Dordrecht, Springer, pp. 31-55.
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