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Matrix versions of the Hellinger distance

Rajendra Bhatia 1 Stéphane Gaubert 2, 3 Tanvi Jain 4 
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\operatorname{tr}\mathcal{A}(A,B)-\operatorname{tr}\mathcal{G}(A,B)\right]^{1/2},$ 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 $\|A^{1/2}-B^{1/2}\|_2,$ 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 $d(A,B)$ 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.
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Submitted on : Tuesday, January 8, 2019 - 3:09:36 PM
Last modification on : Friday, February 4, 2022 - 3:23:32 AM

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Rajendra Bhatia, Stéphane Gaubert, Tanvi Jain. Matrix versions of the Hellinger distance. Letters in Mathematical Physics, Springer Verlag, 2019, 109, pp.1777-1804. ⟨10.1007/s11005-019-01156-0⟩. ⟨hal-01973935⟩



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