A Novel Approach to Canonical Divergences within Information Geometry

• Main Authors: Nihat Ay, Shun-ichi Amari
• Format: Article
• Language: English
• Published: MDPI AG 2015-12-01
• Series: Entropy
• Subjects: information geometry; canonical divergence; relative entropy; α-divergence; α-geodesic; duality; geodesic projection
• Online Access: http://www.mdpi.com/1099-4300/17/12/7866

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.