Standard Divergence in Manifold of Dual Affine Connections

  • Shun-ichi AmariEmail author
  • Nihat Ay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)


A divergence function defines a Riemannian metric G and dually coupled affine connections \(\left( \nabla , \nabla ^{*}\right) \) with respect to it in a manifold M. When M is dually flat, a canonical divergence is known, which is uniquely determined from \(\left\{ G, \nabla , \nabla ^{*}\right\} \). We search for a standard divergence for a general non-flat M. It is introduced by the magnitude of the inverse exponential map, where \(\alpha =-(1/3)\) connection plays a fundamental role. The standard divergence is different from the canonical divergence.


Divergence Function Canonical Divergence Standard Divergence Affine Connection Pythagorean Theorem 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.RIKEN Brain Science InstituteWako-shi, SaitamaJapan
  2. 2.Max-Planck Institute for Mathematics in ScienceLeipzigGermany

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