A Novel Approach to Canonical Divergences within Information Geometry

A divergence function defines a Riemannian metric g and dually coupled affine connections ∇ and ∇^∗ with respect to it in a manifold M. When M is dually flat, that is flat with respect to ∇ and ∇^∗, a canonical divergence is known, which is uniquely determined from (M,g,∇,∇^∗). 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.