A Novel Approach to Canonical Divergences within Information Geometry

Nihat Ay and Shun-ichi Amari

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Submission date: 28. Oct. 2015
Pages: 26
published in: Entropy, 17 (2015) 12, p. 8111-8129 
DOI number (of the published article): 10.3390/e17127866
Bibtex
Keywords and phrases: information geometry, canonical divergence, relative entropy, α-divergence, α-geodesics, duality, geodesic projection
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Abstract:
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.