A novel approach to canonical divergences within information geometry

Abstract

A divergence function on a manifold M defines a Riemannian metric g and dually coupled affine connections ∇ and ∇* on 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.