Consistent estimators of a new class of covariance matrix distances in the large dimensional regime

The problem of estimating the distance between two covariance matrices is considered. A general estimator is provided for a class of metrics, the estimator of which has never been addressed before in the literature. This corresponds to distances that can be expressed as sums of traces of functions that are separately applied to each covariance, which is the case of multiple covariance distances recently derived by Riemannian geometry considerations. The proposed family of estimators is shown to be consistent when both the sample size and the observation dimension increase to infinity at the same rate. In particular, a closed form expression is derived for an estimator of the log-Euclidean metric between covariance matrices. Numerical evaluations demonstrate the effectiveness of this estimator even in relatively small dimensional settings.