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Geometry on Positive Definite Matrices Induced from V-Potential Function

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Geometric Science of Information (GSI 2013)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8085))

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Abstract

In this paper we investigate the dually flat structure of the space of positive definite matrices induced by the class of convex functions called V-potentials from the viewpoints of information geometry. It is proved that the geometry is invariant under special linear group actions. As an application to statistics, we finally give the correspondence between the obtained geometry on positive definite matrices and the one on elliptical distributions induced from a certain Bregman divergence.

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Author information

Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, University of Fukui, Fukui, 910-8507, Japan

    Atsumi Ohara

  2. Institute for Statistical Mathematics, Tachikawa, Tokyo, 190-0014, Japan

    Shinto Eguchi

Authors
  1. Atsumi Ohara
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  2. Shinto Eguchi
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Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc, Tokyo, Japan

    Frank Nielsen

  2. Thales Land Air Systems, 91470, Limours, France

    Frédéric Barbaresco

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Ohara, A., Eguchi, S. (2013). Geometry on Positive Definite Matrices Induced from V-Potential Function. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_69

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  • DOI: https://doi.org/10.1007/978-3-642-40020-9_69

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40019-3

  • Online ISBN: 978-3-642-40020-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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