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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

A Novel Approach to Canonical Divergences within Information Geometry

Nihat Ay and Shun-ichi Amari


A divergence function defines a Riemannian metric $g$ and dually coupled affine connections $\nabla$ and $\nabla^*$ with respect to it in a manifold $M$. When $M$ is dually flat, that is flat with respect to $\nabla$ and $\nabla^*$, a canonical divergence is known, which is uniquely determined from $(M,g, \nabla, \nabla^*)$. We propose a natural definition of a canonical divergence for a general, not necessarily flat, $M$ by using the geodesic integration of the inverse exponential map. The new definition of a canonical divergence reduces to the known canonical divergence in the case of dual flatness. Finally, we show that the integrability of the inverse exponential map implies the geodesic projection property.

information geometry, canonical divergence, relative entropy, α-divergence, α-geodesics, duality, geodesic projection

Related publications

2015 Journal Open Access
Nihat Ay and Shun'ichi Amari

A novel approach to canonical divergences within information geometry

In: Entropy, 17 (2015) 12, pp. 8111-8129