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Logarithmic Divergences: Geometry and Interpretation of Curvature

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Geometric Science of Information (GSI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11712))

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Abstract

We study the logarithmic \(L^{(\alpha )}\)-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent to Kurose’s geometric divergence. In particular, the \(L^{(\alpha )}\)-divergence is a canonical divergence of a statistical manifold with constant sectional curvature \(-\alpha \). For such a manifold, we give a geometric interpretation of its sectional curvature in terms of how the divergence between a pair of primal and dual geodesics differ from the dually flat case. Further results can be found in our follow-up paper [27] which uncovers a novel relation between optimal transport and information geometry.

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Author information

Authors and Affiliations

  1. Department of Statistical Sciences, University of Toronto, Toronto, Canada

    Ting-Kam Leonard Wong

  2. Department of Mathematics, University of Southern California, Los Angeles, USA

    Jiaowen Yang

Authors
  1. Ting-Kam Leonard Wong
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  2. Jiaowen Yang
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Corresponding author

Correspondence to Ting-Kam Leonard Wong .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories, Inc., Tokyo, Japan

    Frank Nielsen

  2. Thales, Limours, France

    Frédéric Barbaresco

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Wong, TK.L., Yang, J. (2019). Logarithmic Divergences: Geometry and Interpretation of Curvature. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_43

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  • DOI: https://doi.org/10.1007/978-3-030-26980-7_43

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26979-1

  • Online ISBN: 978-3-030-26980-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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