Computational Information Geometry Wonderland

Dually Flat Manifolds and Global Information Geometry,, Nihat Ay and Wilderich Tuschmann, Preprint 24, MPI Math Leipzig, 2002. Appeared also in Open Systems & Information Dynamics, Volume 9 , Issue 2 (2002) t

Pages: 195 - 200
Year of Publication: 2002 ISSN:1230-1612

The authors rise and solve the following question:
Does any Riemannian manifold (M,g) admits a dually flat structure on M (ie, a pair of dually torsion-free connections) ?
The answer is no (eg., compact Riemannian manifolds with finite fundamental group, independent of the metric: this is a purely topological argument). Then the authors go one on finding the conditions for such an existence by considering pair of connections with at least one of them complete, meaning that the geodesics are defined on the whole real line. In probabilistic information geometry, it is the case for the exponentia connection nabla^(e) but the mixture connection nabla^(m) is not complete creating a full range of optimization problems involving Bregman projections. Finally, applications to quantum information geometry is reported. This is a well-written paper that focuses on global topological property of dual flatness, a must-read that I recommend.