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A Riemannian Geometry in the \(q\)-Exponential Banach Manifold Induced by \(q\)-Divergences

  • Héctor R. QuicenoEmail author
  • Gabriel I. Loaiza
  • Juan C. Arango
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

In this chapter we consider a deformation of the nonparametric exponential statistical models, using the Tsalli’s deformed exponentials, to construct a Banach manifold modelled on spaces of essentially bounded random variables. As a result of the construction, this manifold recovers the exponential manifold given by Pistone and Sempi up to continuous embeddings on the modeling space. The \(q\)-divergence functional plays two important roles on the manifold; on one hand, the coordinate mappings are in terms of the \(q\)-divergence functional; on the other hand, this functional induces a Riemannian geometry for which the Amari’s \(\alpha \)-connections and the Levi-Civita connections appears as special cases of the \(q\)-connections induced, \(\bigtriangledown ^{(q)}\). The main result is the flatness (zero curvature) of the manifold.

Keywords

Fisher Information Orlicz Space Exponential Family Bayesian Estimator Geodesic Curve 
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.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Héctor R. Quiceno
    • 1
    Email author
  • Gabriel I. Loaiza
    • 1
  • Juan C. Arango
    • 1
  1. 1.Universidad EafitMedellinColombia, Suramérica

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