Matrix versions of the Hellinger distance

Abstract

On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$ 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 $\mathcal{G}(A,B)=(A^{1/2}B{A}^{1/2}{)}^{1/2}$ it is the Bures-Wasserstein metric. We study two other cases: $\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}B{A}^{-1/2}{)}^{1/2}{A}^{1/2},$ the Pusz-Woronowicz geometric mean, and $\mathcal{G}(A,B)=\exp \left(\frac{\log A+\log B}{2}\right),$ 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.

Theorem 9 has been corrected