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Entropy 2015, 17(12), 8111-8129; doi:10.3390/e17127866

A Novel Approach to Canonical Divergences within Information Geometry

1
Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, Leipzig 04103 , Germany
2
Faculty of Mathematics and Computer Science, University of Leipzig, PF 100920, Leipzig 04009, Germany
3
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
4
Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako-shi Hirosawa 2-1, Saitama 351-0198, Japan
*
Author to whom correspondence should be addressed.
Academic Editor: Kevin H. Knuth
Received: 12 October 2015 / Revised: 21 November 2015 / Accepted: 25 November 2015 / Published: 9 December 2015
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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. View Full-Text
Keywords: information geometry; canonical divergence; relative entropy; α-divergence; α-geodesic; duality; geodesic projection information geometry; canonical divergence; relative entropy; α-divergence; α-geodesic; duality; geodesic projection
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Ay, N.; Amari, S.-I. A Novel Approach to Canonical Divergences within Information Geometry. Entropy 2015, 17, 8111-8129.

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