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Skew Jensen-Bregman Voronoi Diagrams

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Transactions on Computational Science XIV

Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 6970))

Abstract

A Jensen-Bregman divergence is a distortion measure defined by a Jensen convexity gap induced by a strictly convex functional generator. Jensen-Bregman divergences unify the squared Euclidean and Mahalanobis distances with the celebrated information-theoretic Jensen-Shannon divergence, and can further be skewed to include Bregman divergences in limit cases. We study the geometric properties and combinatorial complexities of both the Voronoi diagrams and the centroidal Voronoi diagrams induced by such as class of divergences. We show that Jensen-Bregman divergences occur in two contexts: (1) when symmetrizing Bregman divergences, and (2) when computing the Bhattacharyya distances of statistical distributions. Since the Bhattacharyya distance of popular parametric exponential family distributions in statistics can be computed equivalently as Jensen-Bregman divergences, these skew Jensen-Bregman Voronoi diagrams allow one to define a novel family of statistical Voronoi diagrams.

This journal article revises and extends the conference paper [1] presented at the International Symposium on Voronoi Diagrams (ISVD) 2010. This paper includes novel extensions to matrix-based Jensen-Bregman divergences, and present the general framework of skew Jensen-Bregman Voronoi diagrams that include Bregman Voronoi diagrams as particular cases. Supporting materials available at http://www.informationgeometry.org/JensenBregman/

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Author information

Authors and Affiliations

  1. École Polytechnique Palaiseau, France

    Frank Nielsen

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

    Frank Nielsen

  3. CEREGMIA-UAG, University of Antilles-Guyane Martinique, France

    Richard Nock

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  1. Frank Nielsen
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  2. Richard Nock
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Editors and Affiliations

  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  2. Exascala Ltd., Unit 9, 97 Rickman Drive, B15 2AL, Birmingham, UK

    C. J. Kenneth Tan

  3. Département des sciences géomatiques, Université Laval Centre de Recherche en Géomatique, G1V 0A6, Québec, QC, Canada

    Mir Abolfazl Mostafavi

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Nielsen, F., Nock, R. (2011). Skew Jensen-Bregman Voronoi Diagrams. In: Gavrilova, M.L., Tan, C.J.K., Mostafavi, M.A. (eds) Transactions on Computational Science XIV. Lecture Notes in Computer Science, vol 6970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25249-5_4

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  • DOI: https://doi.org/10.1007/978-3-642-25249-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25248-8

  • Online ISBN: 978-3-642-25249-5

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