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 . For the latter (continuous) we derive a gradient descent algorithm to project from the ambient space of covariance matrices to wald space and for both we derive computational methods to compute geodesics in polynomial time and show numerically that the two information geometries (discrete and continuous) are very similar.
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 , as we show, equivalent to any metric obtained from an f -divergence . The second geometry is obtained analogously but using a related continuous-valued Gaussian process on each tree, and it can be viewed as the trace metric of the affine-invariant metric for covariance matrices. We derive a gradient descent algorithm to project from the ambient space of covariance matrices to wald space and for both we derive computational methods to compute geodesics in polynomial time and show numerically that the two information geometries (discrete and continuous) are very similar.