A novel approach to canonical divergences within information geometry

Publikationstyp
Journal Article
Date Issued
2015-12-09
Sprache
English
Author(s)
Ay, Nihat  
Amari, Shun'ichi  
TORE-URI
http://hdl.handle.net/11420/14046
Journal
Entropy  
Volume
17
Issue
12
Start Page
8111
End Page
8129
Citation
Entropy 17 (12): 8111-8129 (2015)
Publisher DOI
10.3390/e17127866
Scopus ID
2-s2.0-84952690948
Publisher
MDPI
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.
Subjects
Canonical divergence
Duality
Geodesic projection
Information geometry
Relative entropy
α-divergence
α-geodesic
DDC Class
004: Informatik
510: Mathematik