Preprint 73/2015

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
Keywords and phrases: information geometry, canonical divergence, relative entropy, α-divergence, α-geodesics, duality, geodesic projection
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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.

03.07.2017, 01:42