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Standard Divergence in Manifold of Dual Affine Connections

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

Abstract

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.

Keywords

Divergence Function Canonical Divergence Standard Divergence Affine Connection Pythagorean Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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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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