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Letters in Mathematical Physics

, Volume 109, Issue 8, pp 1777–1804 | Cite as

Matrix versions of the Hellinger distance

  • Rajendra Bhatia
  • Stephane Gaubert
  • Tanvi JainEmail author
Article

Abstract

On the space of positive definite matrices, we consider distance functions of the form \(d(A,B)=\left[ \mathrm{tr}\mathcal {A}(A,B)-\mathrm{tr}\mathcal {G}(A,B)\right] ^{1/2},\) where \(\mathcal {A}(A,B)\) is the arithmetic mean and \(\mathcal {G}(A,B)\) is one of the different versions of the geometric mean. When \(\mathcal {G}(A,B)=A^{1/2}B^{1/2}\) this distance is \(\Vert A^{1/2}-B^{1/2}\Vert _2,\) and when \(\mathcal {G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}\) it is the Bures–Wasserstein metric. We study two other cases: \(\mathcal {G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},\) the Pusz–Woronowicz geometric mean, and \(\mathcal {G}(A,B)=\exp \big (\frac{\log A+\log B}{2}\big ),\) the log Euclidean mean. With these choices, d(AB) is no longer a metric, but it turns out that \(d^2(A,B)\) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy.

Keywords

Geometric mean Matrix divergence Bregman divergence Relative entropy Strict convexity Barycentre 

Mathematics Subject Classification

15B48 49K35 94A17 81P45 

Notes

Acknowledgements

The authors thank F. Hiai and S. Sra for helpful comments and references, and the anonymous referee for a careful reading of the manuscript. The first author is grateful to INRIA and Ecole polytechnique, Palaiseau for visits that facilitated this work, and to CSIR(India) for the award of a Bhatnagar Fellowship.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Ashoka UniversitySonepatIndia
  2. 2.INRIA and CMAPEcole Polytechnique, CNRSPalaiseauFrance
  3. 3.Indian Statistical InstituteNew DelhiIndia

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