# Matrix versions of the Hellinger distance

## TL;DR

This paper introduces and analyzes matrix distance functions based on different geometric means, exploring their properties and applications to barycenter computation in positive definite matrices.

## Contribution

It extends the concept of matrix Hellinger distances by studying new divergence measures derived from various matrix means, including the Pusz-Woronowicz and log Euclidean means.

## Key findings

- Certain divergences are strictly convex functions.
- Characterizations of barycenters with respect to these divergences.
- Connections between these divergences and known metrics like Bures-Wasserstein.

## 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 $\|A^{1/2}-B^{1/2}\|_2,$ 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.