Towards a Canonical Divergence within Information Geometry

In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove 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 propose a novel definition of canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure $(\mathrm{g},\nabla,\nabla^*)$ of a smooth manifold $\mathrm{M}$. Additionally, we show 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. The case (c) leads to a further comparison of the novel divergence with the one introduced by Henmi and Kobayashi.Comment: 68 Pages, 4 figure