Towards a canonical divergence within Information Geometry

In Riemannian geometry geodesics (up to re-parametrization) are integral curves of the Riemannian distance gradient. We extended this classical result to the framework of Information Geometry. In particular, we obtained that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we have recently proposed a new definition of canonical divergence and its dual function. This new divergence allows us to recover a given dualisticstructure of a smooth manifold . Additionally, we showed that this divergence reduces to the canonical divergence proposed by Ay and Amari in the case of: (a) self-duality, (b) dual flatness, (c) statistical geometric analogue of the concept of symmetric spaces in Riemannian geometry. As matter of fact, when computed on finite states, in the classical and the quantum setting, the new divergence turns out to be the well-known-divergence.