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Averaging Symmetric Positive-Definite Matrices

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Handbook of Variational Methods for Nonlinear Geometric Data

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Abstract

Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas, such as medical imaging, radar signal processing, and mechanics. For the purpose of denoising, resampling, clustering or classifying data, it is often of interest to average a collection of symmetric positive definite matrices. This paper reviews and proposes different averaging techniques for symmetric positive definite matrices that are based on Riemannian optimization concepts.

This work was supported by the Fundamental Research Funds for the Central Universities (No. 20720190060).

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Notes

  1. 1.

    The family of Riemanian metrics that satisfy the affine invariance property is described in [34]; see also Sect. 20.5. The Riemannian metric (20.2) is also called the natural metric [31], the trace metric [43], or the Rao–Fisher metric [63].

  2. 2.

    http://bezout.dm.unipi.it/software/mmtoolbox/.

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Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    **nru Yuan & Kyle A. Gallivan

  2. School of Mathematical Sciences, **amen University, **amen City, People’s Republic of China

    Wen Huang

  3. Department of Mathematical Engineering, ICTEAM Institute, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Pierre-Antoine Absil

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Correspondence to Wen Huang .

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  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Yuan, X., Huang, W., Absil, PA., Gallivan, K.A. (2020). Averaging Symmetric Positive-Definite Matrices. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_20

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