Before revision:  The latter can be viewed as the trace metric of the affine-invariant metric for covariance matrices, the former is that of the Hellinger divergence, or  , as we show, equivalent to any metric obtained from an f -divergence , such as the Jensen-Shannon metric .
After revision:  The latter can be viewed as the trace metric of the affine-invariant metric for covariance matrices, the former is that of the Hellinger divergence, or equivalently , as we show, equivalent to any metric obtained from an f -divergence , such as the Jensen-Shannon metric .