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Matrix Information Geometry pp 111–122Cite as

Jensen Divergence-Based Means of SPD Matrices

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Abstract

Computing matrix means is becoming more and more important in modern signal processing involving processing of matrix-valued images. In this communication, we define the mean for a set of symmetric positive definite (SPD) matrices with respect to information-theoretic divergences as the unique minimizer of the average divergence, and compare it with the means computed using the Riemannian and Log-Euclidean metrics respectively. For the class of divergences induced by the convexity gap of a matrix functional, we present a fast iterative concave-convex optimization scheme with guaranteed convergence to efficiently approximate those divergence-based means.

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Notes

  1. 1.

    We term this trend maxel imaging. Maxel stands for MAtrix \(\times \) ELement imaging by analogy to pixel (PIcture \(\times \) ELement) and voxel (Volume \(\times \) ELement).

  2. 2.

    http://math.nist.gov/javanumerics/jama/

  3. 3.

    www.informationgeometry.org/SPD/

  4. 4.

    www.informationgeometry.org/SPD/

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Acknowledgments

This work was partly supported by the Ecole Polytechnique, Sony Computer Science Laboratories Inc, and the Indo-French Centre for the Promotion of Advanced Research (IFCPAR) to Nielsen, the NIH grant NS066340 to Vemuri, and the University of Florida Allumni Fellowship to Liu. The author warmly thank Professor Rajendra Batia and all participants of the workshop on Matrix Information Geometries for their valuable feedbacks.

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Authors and Affiliations

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

    Frank Nielsen

  2. Department of CISE, University of Florida, Gainesville, FL, USA

    Meizhu Liu & Baba C. Vemuri

Authors
  1. Frank Nielsen
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  2. Meizhu Liu
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  3. Baba C. Vemuri
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Corresponding author

Correspondence to Frank Nielsen .

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Editors and Affiliations

  1. Higashi Gotanda 3-14-13, Shinagawa-Ku, 141-0022, Japan

    Frank Nielsen

  2. S.J.S. Sansanwal Marg 7, Delhi, 110016, India

    Rajendra Bhatia

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Nielsen, F., Liu, M., Vemuri, B.C. (2013). Jensen Divergence-Based Means of SPD Matrices. In: Nielsen, F., Bhatia, R. (eds) Matrix Information Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30232-9_6

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  • DOI: https://doi.org/10.1007/978-3-642-30232-9_6

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  • Print ISBN: 978-3-642-30231-2

  • Online ISBN: 978-3-642-30232-9

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