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Articles

Matrix versions of the Hellinger distance

Published in Springer

2019

On the space of positive definite matrices, we consider distance functions of the form π(π΄,π΅)=[trπ(π΄,π΅)βtrπ’(π΄,π΅)]1/2,$d(A,B)={[\mathrm{t}\mathrm{r}\mathcal{A}(A,B)\beta \x88\x92\mathrm{t}\mathrm{r}\mathcal{G}(A,B)]}^{1/2},$ where π(π΄,π΅)$\mathcal{A}(A,B)$ is the arithmetic mean and π’(π΄,π΅)$\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When π’(π΄,π΅)=π΄1/2π΅1/2$\mathcal{G}(A,B)={A}^{1/2}{B}^{1/2}$ this distance is βπ΄1/2βπ΅1/2β2,$\beta \x80\x96{A}^{1/2}\beta \x88\x92{B}^{1/2}{\beta \x80\x96}_{2},$ and when π’(π΄,π΅)=(π΄1/2π΅π΄1/2)1/2$\mathcal{G}(A,B)=({A}^{1/2}B{A}^{1/2}{)}^{1/2}$ it is the BuresβWasserstein metric. We study two other cases: π’(π΄,π΅)=π΄1/2(π΄β1/2π΅π΄β1/2)1/2π΄1/2,$\mathcal{G}(A,B)={A}^{1/2}({A}^{\beta \x88\x921/2}B{A}^{\beta \x88\x921/2}{)}^{1/2}{A}^{1/2},$ the PuszβWoronowicz geometric mean, and π’(π΄,π΅)=exp(logπ΄+logπ΅2),$\mathcal{G}(A,B)=\mathrm{exp}\beta \x81\u2018{\textstyle (}\frac{\mathrm{log}\beta \x81\u2018A+\mathrm{log}\beta \x81\u2018B}{2}{\textstyle )},$ the log Euclidean mean. With these choices, *d*(*A*, *B*) is no longer a metric, but it turns out that π2(π΄,π΅)${d}^{2}(A,B)$ 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.

Postprint Version

Content may be subject to copyright.This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Spr... ...This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Natureβs AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s11005-019-01156-0

About the journal

Journal | Data powered by TypesetLetters in Mathematical Physics |
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Publisher | Data powered by TypesetSpringer |

Open Access | No |